When Iannucci was approached to make the film, he was busy shooting Veep: "But I read the book, which was horrifying and funny, and had all the themes I was interested in, like how does a dictator run a country through sheer force of personality, how are people terrorised by one man?" A year later, he decided to leave Veep to carry on without him, putting it in the hands of a new show runner: "I didn't want to hang around like Pope Benedict," he said at the time, "getting old in another room while he tried to push through with his reforms." He was also fed up with commuting between the US and the UK, leaving his wife, Rachael, to run their home and family for four months of the year. They have been together since meeting at university, and have three children, the youngest 14 and the eldest, 23-year-old Emilio, an actor. "He pops up in the film playing a young doctor," says his father. "He gets to cut Stalin's head off." The film is a departure for Iannucci, a period piece with a cast he had not worked with before: "I like being outside my comfort zone," he tells me. That zone covers a lot of territory: as well as the political satire he is best known for, he writes columns for the classical music magazine Gramophone – he recently learnt to play the piano, taking his Grade One music exam along with a bunch of five-year-olds – and in 2009 made a documentary about Milton's Paradise Lost. A scene shows him running up and down London's Tower Bridge with a picture of the poet asking people if they'd heard of John Milton: only one person had and she was a visiting Australian. Iannucci was born in Glasgow to a Scottish mother and Italian father. He went to a Jesuit school, St Aloysius, where he was famed for his comic turns. "I was also quite academic at school," he says, "winning all the prizes, which pleased my Dad, though he didn't know I was harbouring an ambition to be a comedy writer. He died when I was 17." After studying English at Oxford University, Iannucci stayed on to do a PhD but abandoned it after he was spotted performing and offered a job with BBC Radio Scotland. A move to BBC headquarters in London followed and in 1991 he produced his first show, a spoof news program called On the Hour. His long-term writing partner, David Schneider, was a fellow student at Oxford: "Arm was not cool," he told an interviewer in 2012. "He listened to Mahler rather than The Clash and dressed as if he was already 40." As a Scots-Italian Catholic from Glasgow, said Schneider, "Arm had the marginal quality of the slight outsider: the best comics need to feel slightly bullied."

Iannucci went on to create a series and film featuring fictional TV personality Alan Partridge, played by Steve Coogan, a wincingly un-self-aware character – like The Office's David Brent – who could so easily and frighteningly be real. After fronting his own show on Channel 4, Iannucci returned to the BBC in 2005 to make The Thick of It, described by political commentator Andrew Marr as "the angry, rampaging bastard child of Yes Minister", and featuring the heroically potty-mouthed Malcolm Tucker, widely assumed to be based on Tony Blair's spin doctor, Alastair Campbell. Martin Sixsmith worked for the Blair government and later advised on the internal workings of the civil service for The Thick of It. Iannucci was a modern-day Molière, Sixsmith told an interviewer, "with a sharp eye for hypocrisy". David Schneider observed that Iannucci's comedy "was fuelled by a quiet moral rage" Iannucci says Blair did some good stuff as Britain's PM: "He left health, education and other things in good shape. But then he blows it [by invading] Iraq despite millions of people protesting and every expert in the UK and the world saying it would be a disaster, telling himself that the politics of conviction are nobler than practical politics. He said this thing afterwards: 'Judgments aren't the same as facts ... I only know what I believe.' His is the faith thing, which is fine if you want to go into a monastery but not if you are running a country and in charge of people who might get shot." It's difficult just to be funny about Donald; he's not an idiot, he has an element of genius but also a dangerous instability and narcissism. He's not a clown and he can do a lot of damage. David Schneider, writing partner The two first met before Blair was prime minister: "He wanted to be interviewed by Alan Partridge so we wrote a funny script for him. He turned up 10 minutes before the show, not having seen the script, read over it and then went out and did it all word perfect – natural and relaxed with impeccable timing. That's when I thought, 'There is something deeply worrying about this.' "

A few days before we meet, Blair has popped up urging Britons to rise up against Brexit and work to change the minds of those who voted last year to leave the European Union. What did Iannucci make of this? "Blair is so discredited," he says, "and [his speech] was all part of this notion that the Leavers got it wrong. I voted Remain, but we lost, so there's no point trying to change the result. A whole section of the population felt ignored and not listened to, so how can you turn around and tell them they got it wrong?" Which brings us inevitably to Donald Trump: "In his [presidential] campaign he had the instinct to hone in on the lurking fears of voters," says Iannucci. "He was very good at showing that he got it." There is nothing so seductive as feeling understood? "Yes, exactly, and they're not even looking for the perfect solution, they just want someone to acknowledge their issues." Iannucci has been quite obsessed with Trump of late, tweeting and re-tweeting about him and penning an open letter to him in The Guardian: "You confuse us," he wrote. "We want to laugh at your stumbles but are petrified by what those stumbles may lead to." Can we expect a satirical show about Trump from him? He shakes his head. "I don't think any fictional attempt will ever match what he's going to do. The best thing so far was McCarthy doing Sean Spicer: it was a crystallising moment." (Comedian Melissa McCarthy has made spoof appearances as the White House press secretary on US show Saturday Night Live.) "What has Trump done in response? Just banned media from press conferences. It's difficult just to be funny about Donald; he's not an idiot, he has an element of genius but also a dangerous instability and narcissism. He's not a clown and he can do a lot of damage." Tony Schwartz, the journalist who ghosted Trump's autobiography, The Art of the Deal, said that Trump was easily bored and had almost zero concentration. Will he get bored with being the leader of the free world? "He'll either get bored," says Iannucci, "or the crossing point will be when his own base starts criticising him and he'll respond by attacking them – he won't be able to help himself. Then, when he's sacked everyone, he'll have no one else to blame. It took Nixon six or seven years to reach that point. Trump has got there in about seven weeks."

For his new film Credit:Raven & Snow Several people I talked to for this story described Iannucci as a genius, capable of holding several ideas in his brain at the same time. Certainly, though he is a perfectly amiable interviewee, I feel that my questions are taking up only a small amount of his bandwidth. As we tuck into our lunch – Iannucci has ox cheek – a few streets away the Independent Inquiry into Child Sexual Abuse is holding its first day of public hearings. On the stand are survivors of the forced migration of children from Britain to Australia after World War II. Does Iannucci think anything, such as child sex abuse, is off-limits for comedy? "Well, it's not," he says, "but the question after that is, how good is the comedy? If you're going to take on an emotionally loaded subject, obviously you have to do it well. I've never been one of those who say you can say anything you like. That's lazy. The most provocative comedy has a set of rules placed on it; when you test those rules, that's when it feels dangerous and exciting." I'm not sure I've seen jokes about child sex abuse, I say. Iannucci counters by noting that fellow satirist Chris Morris covered it in the British TV show Brass Eye. (The 2001 special episode titled Paedogeddon received a record number of complaints, including from many who hadn't seen it.) "It satirised the hysteria around paedophilia," says Iannucci, adding that there was no shortage of material: "A real-life paediatrician had her name plate daubed with insults." Lunch over, we repair to an editing suite to watch scenes from The Death of Stalin, which Iannucci hopes I will find funny. I do. In one scene, a Shostakovich concert is being broadcast live on Radio Moscow. As it ends, a message comes from Stalin: he would like a copy of it sent over. But the concert was not recorded. The ensuing panic is both hilarious and indicative of the terror exerted by the diminutive Soviet dictator.

Iannucci has assembled a cast with serious acting chops – "I don't like working with arseholes," he says – including Steve Buscemi, Michael Palin and Rupert Friend, who all speak in their own vernacular. (Stalin sounds like the general secretary of a mineworkers' union.) "You feel more 'there' than if they were putting on fake Russian accents," says Iannucci. His writing method, he explains, is to gather with partners David Schneider and Ian Martin to brainstorm a script, which someone writes up for the next session, a process that is repeated as often as he considers necessary. "Anything with Arm involves an enormous amount of rewriting," Ian Martin says. "Everyone rewrites everyone else's stuff. We rewrite on set, it's relentless, and we end up with this sort of ghost forest of abandoned lines and words." Iannucci is in control of everything, says Martin: "He is a genius, and always the cleverest person in the room." He pauses, "Unless Chris Morris is also in the room, then Arm is only just the cleverest." The Thick of It was a coruscatingly funny send-up of Westminster, but Iannucci is also an astute political commentator. He has been a stern critic of what he sees as attempts by the Conservative government to muzzle the BBC. "The same thing happens with the ABC in Australia and the CBC in Canada," he says, "these conservative administrations seeing the national broadcaster as a liberal-left mouthpiece and threatening to do something about it by pruning it. Just as many people complain about the BBC being right wing as those who say it has a left-wing bias, so it must be doing something right. And it's such an international brand, the government should be promoting it as a major export that you can make tons of money from. Instead, they say, 'Let's shut it down a bit.' When the BBC's charter was up for renewal, the then culture secretary John Whittingdale invited Iannucci "for a chat". Did he go? "I did, and to be fair he accepted that the government hadn't been listening to the creative industries." Iannucci rounded up people from drama, arts and sport to meet the minister and his team: "It did lead to a [charter renewal] White Paper that was a lot less hostile. So we felt we'd done a bit of good."

Loading Iannucci is coming to Australia for the Sydney Writers' Festival, and bringing his family with him to visit his wife's cousin and other friends in Adelaide. Then he will be back to work on his next project, a film of David Copperfield. "I'm a huge Dickens fan," he says. "It will be set in Victorian times, and we'll use as much of Dickens' dialogue as possible but with a contemporary cast using their own accents, like in Stalin." He is also working on a pilot for a new HBO show. "It's a sci-fi comedy, set a bit in the future, mostly in space. That's about it for the moment." Armando Iannucci will appear at the Sydney Writers' Festival at Sydney Town Hall on May 2, and and is presented by the Wheeler Centre at the Comedy Theatre, Melbourne, on May 3.