Twelve months ago, only fans of the BBC show Wallander or maybe those who saw him in Joanna Hogg's film Unrelated would've picked Tom Hiddleston out of a crowd. The past year has rather changed matters. Rarely, if ever, has a British actor made such an impression so swiftly – working back-to-back with Kenneth Branagh, Woody Allen, Terence Davies and now Steven Spielberg, he has not so much made a breakthrough as smashed his way into the public consciousness with all the force of a wrecking ball.

Even he can't quite fathom it. Recently he went back to Rada, where he trained. "Some of the staff wanted me to come back and give dispatches from the front line, to give an illustration of what students might expect on the other side of their training," he explains. "I sat in this room where I'd practised sword-fights and sonnets and Stanislavsky and it felt like I was there yesterday. And I remember saying, 'I'm supposed to be sitting in the chairs where you are, listening to Mike Leigh or Michael Sheen or whomever.' I couldn't believe I was the person giving the talk." k

We first meet in a private members' club in Covent Garden, where Hiddleston and I have been assigned use of the fourth-floor library. Seated on an orange plastic chair, he's tall, thin and handsome in an angular sort of way. His eyes are a fierce blue and his forehead high, supporting a mop of tight brown curls. The handshake is firm, the dress sense preppy – brown brogues, black jeans, and a navy blazer fitted over a white T-shirt. With his Eton education, he seems quintessentially English, though his father was born in Greenock. "When Scotland played England [at rugby], I would support Scotland with him," he confides.

Given that he turns 31 next month, it's not that Hiddleston is particularly young to achieve his successes. It's just the rapidity of his rise that is so shocking. Much of it can be contributed to Branagh. After Hiddleston played his fellow detective in Wallander, and acted alongside him in a West End production of Chekhov's Ivanov, Branagh wanted his co-star for Thor, the 2011 Marvel Comics blockbuster that he (rather surprisingly) directed. Hiddleston was earmarked for the role of Loki, the god of mischief in Norse mythology and the film's main antagonist.

He was brought in by Branagh to read for the role, and when the studio executives saw the audition tape, they went wild. "They said, 'Who is that guy? Why haven't we heard of him before?'" It was a question the rest of Hollywood has since been asking. Woody Allen got in quickly, writing him a letter to ask him to play F Scott Fitzgerald in his delightful fantasy Midnight in Paris. "I didn't audition," says Hiddleston. "I didn't even know he was making a film. He just said, 'Dear Tom, here's the script, I'd love you to play the part. We're shooting in Paris in the summer.'"

Also cast in Terence Davies' recent Terence Rattigan adaptation The Deep Blue Sea (playing Rachel Weisz's lover), at the same time, he got the call to meet Spielberg for his new film, the First World War epic War Horse. "I remember going home that night and thinking 'What is happening to my life? How did I get here? I'm about to meet one of my heroes! Don't fuck it up!'" He didn't. By the time they finished, Spielberg leant across the table and offered him the film there and then. "I nearly whooped, wept, laughed and cried." Even his agents were shocked.

Then again, it's not hard to see what entranced Spielberg. Hiddleston must have seemed perfect for the role of the noble British officer, Captain Nicholls. He may not have any military experience, but it's in his blood. His paternal grandfather was in the Royal Artillery, while his mother's father was in the Navy. His own father, a scientist by trade, once ran a company that supplied artificial limbs to soldiers returning from the Falklands. Then there's his father's great uncle – the last Tom Hiddleston in the family – who was a sergeant in the British Army before he was killed in action, in 1916.

"I always found the extraordinary loss of life in the First World War very moving," he says. "I remember learning about it as a very young child, as an eight- or nine-year-old, asking my teachers what poppies were for. Every year the teachers would suddenly wear these red paper flowers in their lapels, and I would say 'What does that mean?' And in history, the next thing you learnt was that there was this terrible, terrible war, from 1914 to 1918, when the country lost an entire generation of young men. And I remember that really affecting me at a very young age."

As a child, Hiddleston's "musical instrument of choice" was the trumpet. When he was 12, he was selected at school to play the Last Post on Remembrance Sunday. "I remember feeling the weight of that," he says. "That I was heralding the two minutes' silence." He even told Spielberg that "Americans don't really understand" the British attitude to the Great War. "It was quite a European war until 1917, when the Americans joined up. They don't have the same sense of the loss of innocence and the cataclysmic loss of life. A whole generation was wiped out."

Though on one level an old-fashioned family film, War Horse doesn't shrink from the reality, with jaw-dropping battle scenes every bit as shocking as Spielberg's own Saving Private Ryan. Based on the novel by Michael Morpurgo (already the inspiration for a hit play), the story follows a horse bought by a Devon farmer (played by Peter Mullan) and raised by his son Albert (Jeremy Irvine), who names him Joey. When war breaks, the steed is sold to Captain Nicholls – "this agent of separation", as Hiddleston calls him, "who divorces the horse from his boy".

Hiddleston plays Nicholls with just the right level of nobility, promising to get Joey back to Albert when the war ends. "Other writers might have made him quite bluff, disciplinarian and possibly cruel. But Michael Morpurgo makes him kindly and decent, upstanding and modest, which I found very moving. That was one of the things that attracted me to the part. Having dug around in so much damage as Loki, here was a man with such a sensitive soul who found himself in uniform, and fighting on the front line."

Hiddleston will be returning to the "wounded, troubled pain and volatility" of Loki this summer. After watching Thor emerge as one of last summer's biggest hits (it took $450m globally), he is now the lead villain in The Avengers – which, if you're a fan of the Marvel Comics universe at least, will be the blockbuster movie to end them all. It pits the all-conquering Loki against the titular posse of superheroes, gathered together from such recent smash-hit films as Iron Man, Captain America and, indeed, Thor.

Not surprisingly, given the exalted company that sort of get-together entails, Hiddleston calls the experience "entirely surreal". "There was one day when, as Loki, I was sat on some steps, staring at the Avengers in front of me. It would be quite odd just to look at a grouping that involves [such actors as] Robert Downey Jr, Scarlett Johansson, Chris Evans, Samuel L Jackson, Mark Ruffalo, Jeremy Renner and Chris Hemsworth. But the fact that they were dressed in the most outlandish hero costumes was bizarre! And I thought to myself, 'This is the dizzy heights.'"

Still, he must be getting used to this state of disorientation after the year he's just had. Ask him to put it into words, and he can only talk in clichés. "It all happened so quickly, I'm just concentrating on putting one foot in front of the other," he tells me. What about fame? Is he worried about losing his anonymity? "I'm trying to cross every bridge when I come to it. I think it would be very dangerous to start planning the date at which I will be unable to go to Sainsbury's."

Fortunately, he has a stable upbringing to keep those feet anchored. Born in London, Hiddleston grew up in Wimbledon, the middle child of three (he has an older and younger sister). When he was 10, his parents moved to Oxford, after his father James won a job as managing director of a pharmaceutical biotechnology company with links to the university. "He was really stimulated by being an executive go-between, between these two worlds."

Before Hiddleston's mother got pregnant with his elder sister and gave up her work to concentrate on parenthood, she was an arts administrator and casting director for an opera company, having trained as a stage manager. "When I was in my teens, she was always one to suggest possibly going to the Picturehouse [arthouse cinema] rather than the Odeon," he says. "So she was always peppering my appetite for Jurassic Park and Terminator 2 with an appetite for [Michael Winterbottom's] Jude or whatever."

Given this comfortable environment, it's tempting to look for autobiographical details in Hiddleston's work. In particular, Archipelago, his second film for Joanna Hogg, a story of upper-middle-class upheaval – or as he puts it, "the implicit tensions and struggles that happen within every family". His subtle, painful turn as the aimless Edward is arguably his best to date, as the director mined him for information on what it's like to be a young man of his age. "Joanna asked direct questions: what keeps you up at night? What nags at the corners of your soul in darker moments?"

He may well have thought back to his own family discord, when his parents divorced when he was 12. "It was very difficult and I always say that it made me who I am," he told one interviewer last year, "because it made me take responsibility for my life and I saw my parents for the first time as human beings, not as perfect love machines. They were both very badly hurt. I mean, it's hard enough when you're ending a short-term relationship, isn't it? I can't imagine what it's like to end a 17-year marriage. But I'm so proud of them and I couldn't do without them and as a result [of the divorce] I have grown-up, intimate relationships with both of them."

Talking of relationships, there is some confusion I want to clear up with Hiddleston. In the past, it's been erroneously reported that Hiddleston was secretly married to the actress Susannah Fielding. Already featured in FHM last year, the Hampshire-raised 26-year-old made one of her first screen appearances in an episode of Wallander with Hiddleston back in 2008. Just checking, I say, but are you? "I am definitively not married. That was a big mistake. Don't know where it came from." But you are in a relationship right now? "Well, erm, I dunno actually," he says, shyly. "That's kind of an awkward answer, isn't it?"

We leave it at that – after all, Hiddleston is not the sort you find kicking back in the centre pages of OK! magazine, even if his ever-growing circle of female admirers call themselves "Hiddlestoners".

It was during his school days at Eton that he discovered acting, which he then continued at Cambridge. Reading Classics at Pembroke – the same college that legendary comic Peter Cook went to, he says proudly – he joined the Amateur Dramatic Club. "I love the acting community at Cambridge," he says. "It's really quite committed and serious, since the days of Derek Jacobi and Ian McKellen right through to Emma Thompson and Hugh Laurie."

From Eton to Cambridge and on to Rada sounds as traditional an English education as you could get, I suggest. "In some ways it does and in some ways it doesn't. Since my education, I've done quite untraditional things. There are very few Etonians who went to Rada. And far fewer Etonians – certainly when I was there – went to Cambridge. I don't know whether it's the same now. Most people I knew went to Oxford, because it seemed more of an easy bridge."

Arriving at the time of the Laura Spence affair – the high-flying state-school pupil who was refused a place at Oxford, allegedly because of her working-class background – he says the entrance examiners put him through his paces. "They made sure I was able to think for myself and stand on my own two feet, intellectually. Cambridge is a meritocratic place. I know this sounds odd, but I met more kinds of people there than I've probably met in any place in my life. It seemed to be so international, and there were people from all walks of life, all backgrounds. Any chips on any shoulders had to be very swiftly removed, in any direction."

When we speak for a second time, just before Christmas, Hiddleston has been "knee-deep in mud, blood and warrior poetry", playing the title role in a new Sam Mendes-produced BBC film of Shakespeare's Henry V. With plans to shoot Henry IV Part 1 and 2 in early 2012 – in which he will reprise the role as Prince Hal – it truly feels as though he's taking over from his fellow Rada alumnus Branagh, who famously directed himself in his 1989 film of Henry V. Already dubbed "Branagh's boy" in the press, Hiddleston clearly sees him as a mentor. "I trust him implicitly," he says.

I ask whether he chatted with Branagh about taking on the role of Henry V. "Over the years, we've talked about it," he admits. "He was very kind. He sent me an email when he found out. He just said, 'I heard about the Henrys. The very, very best of luck. You'll have a fantastic time.' I was going to ask him about it, then I thought, 'I don't know.' What I took from that email was that I had his blessing, in a sense. And I do feel Shakespeare is like an Olympic torch that gets passed on from generation to generation."

Never mind the Bard: Hiddleston has taken the torch from Branagh, it seems, as he goes once more unto the breach.