Just two hours from Britain is Europe's last dictatorship, a country that recalls Stalinist Russia and where critics of the government 'disappear'. But thanks to an underground theatre group – and supporters including Jude Law and Tom Stoppard – the world is finally waking up to its plight

Britain's greatest living playwright is nodding gently in his sleep. Tom Stoppard is uncomfortably folded into a second-class Eurostar seat, his head lolling against his shoulder. I've been told that he'll give me an interview half an hour before we reach Brussels, so after a moment's hesitation, I tap him on the arm and, like a lizard, he slowly opens one bloodshot eye.

He's not just tired – he's jet-lagged. He's feeling a bit ill, or "odd" as he puts it. He's just returned from America, and has got up at the crack of dawn to catch the Eurostar in order to spend three days attending something called the Brussels Forum. "And I don't even know what the Brussels Forum is," he says. "Do you?"

I shake my head.

"I'm really not entirely sure why I'm here. I was in New York and Natalia rang me up and asked me to come. So I came." He considers this. "Why are you here?"

Natalia asked me, I say, and we share a moment of mutual incomprehension at the fact that he is addressing a conference of Eurocrats and I'm attending one. But then "Natalia", or Natalia Kaliada, is an unstoppable force. Stoppard calls her "the little dynamo", but this is understating the matter. She's the artistic director of the Belarus Free Theatre, previously based in Minsk, now in exile, an organisation that has come to be the central motor of a campaign to draw attention to the terrible political repression happening in a country that, as Natalia points out frequently, to any audience that'll have her, "is in the middle of Europe, only two hours from London".

Five years ago, Natalia wrote to Stoppard and asked him to sign a letter of support, and it's why he now finds himself in situations like this: spending the weekend in Brussels with a hotel full of high-level government ministers, having agreed during an inattentive moment to giving the conference's "opening remarks" before a keynote address by the president of Europe.

"And I really have no idea what to say. What on earth do you think I should talk to them about?" It's a shame that it's only in retrospect that I realise that the man who wrote Rosencrantz and Guildenstern are Dead, and Jumpers, and Arcadia, who was the great champion of the dissident writers of the eastern bloc, is probably not actually seeking my advice. Because, oh dear God, I give it anyway.

You're the only person here who doesn't represent a vested interest, I say. It's really up to you to go and stir things up.

"You think I should bang the Belarus drum?" he says. "I was wondering if I should. Or if I should talk about what the purpose of these forums are?"

Bang the drum! I say. Bang the drum!

In my defence, Natalia Kaliada is a force field I defy anyone to resist. I've watched Jude Law stumble blindly into it. He turned up at the Observer's office on a sunny Saturday, two weeks ago, to introduce her at our TEDxObserver event; next thing you know, he's promised to be at the Houses of Parliament for a performance alongside Natalia's husband, and co-founder of the Belarus Free Theatre, the playwright Nikolai Khalezin.

I, too, find myself pulled along in her wake, following her from event to event and now on to this dawn Eurostar (she's sitting at the back), drawn in by her sheer energy and determination and the fierce commitment that she brings to her cause. Because what's so compelling about what Natalia is doing, and why everyone around her, can't help but be compelled, too, is that, as Jude Law puts it, "she's living it". She and Nikolai and their youngest daughter, 12-year-old sweet-faced Daniella, who hasn't been to school since they fled Minsk, can't go home. Their other daughter, 17-year-old Marya, is still in Belarus, as are their parents and they don't know when they'll ever see them again. They've been named enemies of the state, but, as Natalia repeatedly points out, they're the lucky ones. Many of their friends are in jail; others are dead.

It's almost as if perpetual motion is the only thing that's keeping her going. In between her talk at TEDxObserver and now, she's been to Washington, addressed another conference, and flown back on the redeye. Later, on no sleep, she gave another emotional speech at the Index on Censorship awards, where Tom Stoppard presented her with a special award on behalf of Belarus's prisoners of conscience, as classified by Amnesty, halfway through which she broke down and cried, and now, here on the train, while Stoppard has fallen into a coma and I'm just about managing to stare blankly into space, Natalia is furiously typing letters and emails and texting her contacts in Belarus.

"She doesn't eat, she doesn't sleep," says Irina Bogdanova, a fellow Belarussian whom Natalia and her family are currently staying with in Hampshire. Irina's voice is cracking. She's sucking Strepsils and is rasping away, but then she's another unwilling activist. Another accidental campaigner. "I hate politics!" she says. "Hate it. And yet here I am! It's incredible to me." She's a doctor who's lived in Britain for the past 18 years, her partner is a British aeronautical engineer, but on 19 December last year, her brother, Andrei Sannikov, Belarus's most credible opposition leader, and a presidential candidate, was beaten up, arrested, and is being held in a KGB jail awaiting what in effect amounts to a show trial.

He's one of the prisoners of conscience. And his wife, Irina Khalip, a journalist, is under house arrest with their three-year-old son unable to contact the outside world, and so it's to Amnesty that Irina Bogdanova has found herself turning. "They have been fantastic. Because there is only my 78-year-old mother in Belarus, who's all by herself, who is trying to do it all. We found a lawyer, but they de-barred him, because that's what they do. But I've been speaking to Amnesty's lawyer and they've been advising us. And they act straight away. When something happens, they act so much quicker than anybody else. They issue this urgent action, which means they send a letter and a press release, to all their offices and the government, so at least the government knows that Amnesty is on their case, even if they don't care much. But actually, they do, I think. I mean, I don't think the situation changes drastically, but maybe it doesn't become any worse.

"What Amnesty does is to show that the world is watching. That somebody is taking note. That there are consequences. Or maybe one day, there will be. Because at the moment, nothing is happening. Nobody else is reacting! Can you imagine if it was Britain? If Gordon Brown had been locked up by David Cameron before even the election results had been announced? That is what is happening in Belarus. And this is Europe. Europe!"

There are so many difficulties in writing about what is going on in Belarus. Why you might struggle to find the country on a map. Why you might be struggling to care. There have been so many revolutions this spring, so many populist uprisings, and yet, right here, in Europe, is a forgotten country in the grip of what – you'll discount this as hyperbole and will almost certainly resist believing (I did) – is a Stalinist-style reign of terror.

Or, at least, when I meet Natalia for the first two times – unprepared, in a rush, my mind on other things – I just can't really process what she tells me, can't quite compute it. She's so passionate, so determined to tell her story, to convince me of its rightness, and its urgency, that words such as "torture" and "killing" and "KGB" tumble from her mouth, and I find it hard to piece together a narrative, or understand the context. And while I spent my early 20s hanging around eastern Europe and the ex-Soviet Union, my strongest memory from my last trip to Moscow was the traffic jam to get into Ikea.

And yet talking to Natalia feels like stepping into a DeLorean and speeding back 30 years in time. "Longer," says Irina. "This isn't like Russia in the 80s. It's Russia in 1937. These mass arrests. People being disappeared. It's the purges."

Poor Belarus. It's always been the country in the way of western armies heading east, and eastern armies heading west; a brutal history that has continued into this century, first by Hitler – the Nazis killed a quarter of its population – and then by Stalin. It had the briefest moment of freedom, after the collapse of the Soviet Union, and then in 1994, Alexander Lukashenko, the director of a state farm, was elected president and systematically began stripping those freedoms away again.

And last December, in the worst timing in modern protest history, just months before the uprisings in Tunisia and Egypt, roughly 50,000 Belarusians took to the streets on election night in a peaceful demonstration that was violently put down. Natalia, like many others, was arrested; she, unlike many others, was released (there was a mix-up over her name). According to Amnesty, 42 are still in jail, and every day brings news of fresh trials and increasingly punitive sentences.

Two out of eight presidential candidates are in KGB jails; another is under house arrest, although at one point seven out of the eight had been detained. ("Even in Russia, they renamed the KGB the FSB," says Natalia. "In Belarus, they didn't even bother.") Another, Ales Mikhalevic, has now fled to the Czech Republic and at a press conference told how he was tortured by masked men. "In fact, we've known about torture happening in police cells for years," Heather McGill, Amnesty's country researcher for Belarus, tells me. "This isn't new. None of this is new. It's just got worse. What we're most concerned about, more than these high-profile cases which at least the EU knows about, is that there are an awful lot of young people, who nobody knows about, in jail.

"Yesterday, a 20-year-old law student who was simply peacefully demonstrating was sentenced to three and a half years in a labour camp."

And then there are the "disappeared". When we get to Brussels, I meet another Irina: Irina Krasovskaya, the founder of a movement and website called We Remember. Twelve years ago, her husband, businessman Anatoly Krasovsky, and his friend, Viktor Gonchar, the leader of the opposition, simply disappeared while driving home. Vanished. Today, having told the story hundreds, if not thousands of times, Irina can't keep the emotion out of her voice when she tells me how their bodies have never been found.

It's such an anomaly, Belarus. The last dictatorship in Europe. A regime that, at times, seems to exist as an echo chamber of the past (Irina Bogdanova's nephew, Sannikov and Khalip's son, was removed by the authorities who attempted to place him in a state orphanage, the same tactics employed in Stalin's time). It's even made Tom Stoppard think again about totalitarianism. "Because there were certain assumptions I always made about Czechoslovakia under communism, and I'm now wondering whether those assumptions were actually true. It seemed to me that, there, people were followed and harassed, and they might lose their job or a promotion, the usual menu of that totalitarian system but my impression was that the jails were not full of political dissidents as they are here. The penalties were harsh for the few who did it, but there wasn't this mass fury."

Stoppard was born in Czechoslovakia and became closely involved with the dissident movement there, as well as travelling to the Soviet Union in the 80s with Amnesty, whose value as an organisation he calls "inestimable". What it does, he says, "is connect awareness to protest globally… And, of course, the very phrase 'a prisoner of conscience' is a very potent idea. Someone who's been locked up because of his conscience." But he had no idea that three decades on, having seen the Wall fall, and much of the old eastern bloc become part of the EU, he'd still be fighting the same old battles.

Is what is going on in Belarus now worse than what was happening in the old eastern bloc?

"I don't think there were these extrajudicial killings. The Soviet system was too officious. It was very fond of legalistic paperwork… It was a matter of ideology and obeying it. But Lukashenko doesn't have an ideology, he has a business plan."

A couple of hours later, in the ultra-marble lobby of Brussels's five-star Conrad hotel, I come across a tense grouping of Natalia and the two Irinas. Natalia looks like she's going to cry. Irina Bogdanova has now lost her voice completely. "I am just so tired of all of this!" she says.

They've just found out from a friend in Belarus that 70 more people have been arrested. And when they corner Baroness Ashton, the EU's foreign affairs representative, she gives them the brush-off. "We have responded quickly to the situation in Belarus," she says in the opening session in response to a question that Natalia asked. "I have met with relatives… we have imposed strong sanctions."

Strong sanctions? I ask Natalia afterwards.

"That's bullshit! There are bullshit sanctions. There's a travel ban on some politicians, that's it. We need economic sanctions."

"It's just feels so pointless!" says Irina Bogdanova, while Irina Krasovskaya adds: "Look, Belarus is on the agenda. They mentioned Andrei Sannikov in the opening remarks. Belarus is here."

And it's here because of them. Because, perhaps, most remarkably of all, the fight for freedom is being led by women. "We haven't any choice," says Irina Bogdanova. "I don't want to fight. I don't want to be here doing this. But someone has to. And all the men are in jail." It's the wives, and mothers and sisters and daughters, who've been left to do the dirty work. There's Elena Edwards, sister-in-law of the former presidential candidate Ales Mikhalevic; and Natalia Radina, who's taken over as editor of the main opposition website, Charter 97, after its founder, Oleg Bebenin, was found hanged last September.

And it's why, perhaps, they're so persuasive, so moving. When they talk, it's fuelled by pain and loss and anger. It's not hard to see why Steven Spielberg has become a supporter of the cause, why George Clooney has, why Mick Jagger made a video appeal on their behalf. In fact, in many ways, the Free Belarus Theatre is the model of a modern human rights campaign, although it never set out to be one. Natalia comes from a theatrical family and always wanted to be an actress but after studying in Moscow, she ended up working for NGOs, and Nikolai was a newspaper editor. He edited the three most influential independent newspapers in Belarus, until, one by one, they were all closed down.

What's the main newspaper called now, I ask Natalia at one point. "Sovietskaya Belorussiya. Soviet Belarus! I mean, really, it's beyond a joke! Except it's not funny."

With no newspapers left to work for, Nikolai turned to writing plays, and "one was put on in Moscow and we received royalties and with this money we decided to set up a theatre company with our friend, the director Vladimir Shcherban". It's the company's sixth anniversary this week, and while the authorities take their inspiration from Stalin, they have explicitly borrowed from the samizdat writers of the 1970s and 80s, employing the same methods, putting on plays in private flats and houses.

The British playwright Laura Wade, who went to Minsk to hold workshops with them, and attended one of their underground performances, says that "the thing about them is that not only are they very brave on a personal level, as well as very powerful, but also the work they are doing is incredibly high quality as well".

Being an underground theatre group is one thing; being the figureheads for a campaign to free Belarus is another. But, according to Jude Law, "the theatre has become a symbol of what's happening to the country, and Natalia has become a symbol of the theatre. She's been thrust into the spotlight because she's eloquent and charming, but what they're so good at is getting attention for their work, and then turning that back on to the country."

It's really no surprise to me that after a week or so of intermittently following Natalia Kaliada around, I end up in a room at the House of Commons with her, Jude Law, Kevin Spacey, Tom Stoppard, Sam West, half a dozen MPs, the leading lights of the Young Vic, a brace of playwrights, and a director or two.

"If there's ever the smallest crack," says Stoppard, "then Natalia has her foot in that door." As we wait around, Kevin Spacey, director of the Old Vic, tells me how he saw them in New York when they put on Being Harold Pinter, a play based on Pinter's work, adapted and directed by Vladimir Shcherban. "There was just four actors. And five chairs. It was so powerful, hearing Pinter's words about politics and repression, in their own language. My heart was on the floor."

"I took Harold to see it in Leeds," says Stoppard. "It turned out to be one of his last public outings, and it was tremendously moving."

"I said to them: 'How can I help?'" says Spacey. "And they said: 'We need performance space.' So I said: 'Come to the Vic.' So they're now rehearsing there."

It'd be easy to dismiss them, the luvvies, and the film stars, stepping out of their sparkling celebrity bubbles for just a second, to throw a crumb of human comfort at the exiled and dispossessed, but there's real respect for the work that the Belarus Free Theatre is doing. They've all shown up at a moment's notice. And, crucially, it's a lot more than anybody else is doing – than the British government and EU have done. Tom Stoppard is on his fifth straight day on the Belarus trail. Spacey's not even on the bill. And Jude Law is refreshingly un-A-list about the whole thing. It's hardly a glamour gig: we're in a committee room of the House of Commons, with a handful of MPs, some parliamentary researchers and assorted hangers-on. And yet watching this odd couple – Jude Law in his crisp, white, tailored shirt, Nikolai Khalezin, with his ponytail and earring – do a scratch performance of an extract from Nikolai's play, Generation Jeans, alternately in Russian and English, is genuinely moving, not least because it is, in part, a love letter to Natalia, or "Natasha" as he calls her. (They're inseparable now, but when I ask Natalia how they met, she shudders. "I really hated his ponytail, his leather trousers… ugh! But then, you know, I got to know him.") When he was arrested during the election before last, and was in prison, he dreamed of "green meadows that would not lead me to Natasha", and notes that "when we are at risk our women become samurais".

They do. Poor Irina Bogdanova is in tears again. "It's just too close to me, emotionally," she says. "It's like going through it all again." But Mike Harris, of Index on Censorship, is pleased. "We had a really good turnout of MPs. And Douglas Alexander [the shadow foreign secretary] showed up." Why did they come, do you think? "Because we have tea and celebrities," he says.

And this, in very large part, is down to Natalia. She looks shattered. There are dark circles around her eyes already and she'll be up again at dawn for an interview for BBC World, off next week to New York, answering emails, drumming up support, thinking up schemes. I say goodbye, although not for long. Ping! An email drops into my inbox. "Is there any chance you can help me get a letter to Moby?"

For more information on the campaign, please see freebelarusnow.org. Natalia's talk at TEDxObserver can be watched online at theguardian.com/tedx

This article was amended on 5 April 2011, to correct the attribution of Being Harold Pinter