Wherever Benedict Cumberbatch goes screaming fans follow. They’re mostly female and known in various circles as Cumberbitches, Cumberbabes or, more recently, the Cumber Collective. The premiere of his latest film is no exception.

As the British star steps out of a black limo in a slim-cut 1960s-style suit, the crowd erupts into hysteria akin to Beatlemania. Cumberbatch looks relaxed and calm in the eye of the storm. He takes off his thick-rimmed Persol specs, gives a warm smile and walks towards his adoring public. From selfies to autographs, he is happy to oblige. He’s in true movie-star mode right now.

The Imitation Game is about to be screened as part of the Toronto International Film Festival. Earlier in the day – looking slightly more casual, complete with a five o’clock shadow – he sat down to talk about the film.

These are heady times for Cumberbatch; the buzz surrounding the film predicts an Oscar nod for the London-born star. After shooting to fame on both sides of the Atlantic thanks to BBC TV series Sherlock, he has become one of Hollywood’s hottest properties. With a slew of critically acclaimed movies under his belt, including The Fifth Estate, 12 Years a Slave and August: Osage County, an Academy Award would be the icing on the cake.

Acting was always Cumberbatch’s ambition from an early age, which is just as well, as he may not have fared so well had he chosen a job in the City – he blew his maths GSCE. He laughs when he talks about not reaching his full potential at Harrow School: “I was told that I should have got an A – and I flunked it. I got a B.”

Fortunately, no mathematical qualifications were necessary for his latest role, but they may have helped. Cumberbatch plays genius and code-breaker Alan Turing in The Imitation Game. The film tells the story of how Turing and his team at Bletchley Park helped to solve the Enigma code during the Second World War.

While he was hailed as a hero after the war, Turing’s later life was more complicated. He was charged with gross indecency after admitting to a relationship with a man – an illegal act in the 1950s – and faced hormonal treatment designed to reduce libido. Two years later he committed suicide. Despite his wartime achievements, Turing is still a relatively unknown figure in history. So how much did Cumberbatch know about him before taking on the part?

“I knew a little bit about him from Breaking the Code, the Derek Jacobi televised version of the play,” he says. “But yes, he’s not in the front of history textbooks as he should be and he is remarkably unknown in comparison to how important he is as a figure – political as well as scientific. If the Enigma code hadn’t been cracked maybe Germany would have been victorious and Fascism would have won. And he is the father of the computer, the inventor of the whole concept of the computer, in fact – so a very, very important person.”

Keira Knightley on Cumberbatch: He never takes the easy option. He always throws a bit of a curveball; I love that

Cumberbatch is no stranger to wartime roles, having appeared previously in Cold War thriller Tinker Tailor Soldier Spy and Steven Spielberg’s first world war drama War Horse. But Turing is one of the most complex characters he has tackled to date. So what did his preparation comprise?

“A lot of the heavy lifting is done by Graham [Moore]’s incredible script. It is a remarkable first script, which makes me laugh because I wonder how he’s going to write a second script to match it,” he says, running his fingers through his hair. “But I did the usual forensic scavenging that you do when you’re researching a role. You go to books and biographies, you go to verbal histories of people who knew him, you just dig around and try to apply what makes sense in the prism that you’re looking at him through.”

Cumberbatch says both he and director Morten Tyldum swapped notes from their findings on Turing to ensure as faithful a representation of him as possible, but concessions had to be made for the camera. It was clear from their research that Turing had quite a pronounced stammer, but this had to be toned down so as not to alienate the cinema-going audience. Turing also had an oestrogen implant embedded in his leg as part of his hormonal treatment for homosexuality, which he dug out with a kitchen knife in frustration. This incident isn’t shown or mentioned in the film but Cumberbatch references it with a slight limp.

“There are lots of variables with this man and there were lots of choices to make from moment-to-moment, in his behaviour and interpretations of scenes and the tone of the scenes,” says Cumberbatch, who will do as many takes to get a scene right as time and money will permit. “You only get one chance to do these things. It will be there forever,” he says with a broad smile.

Speaking to Cumberbatch, it’s clear he is a highly professional man and something of a perfectionist, taking a “huge care of duty”, as he puts it, with every role he takes on. His Imitation Game co-star Keira Knightley – who plays fellow code-breaker Joan Clarke – tells us a little more about Cumberbatch at work.

“He’s never frightened of the complex,” says Keira, who first met Cumberbatch when they worked together on Atonement. “He always dives into something and tries to find every single nuance. He’ll search and search and search. And that’s very exciting to work with because he never takes the easy option. He always throws something – a bit of a curveball. And I love that.

“He also puts a lot of pressure on himself and he puts a lot of pressure on everybody, but most of the really great people I’ve worked with do exactly the same,” adds Knightley.

But does working in such an intense way allow Cumberbatch much time for a private life and, if so, what does he get up to when the cameras aren’t rolling? “None of your business,” he laughs. “And I certainly don’t talk about it to journalists,” he says in a slightly disconcerting way. “Sorry I didn’t mean that as aggressively as it came out,” he quickly adds. “That was supposed to be a joke! But yeah, it’s private – and I enjoy that, you know.”

Cumberbatch can be quite intimidating in the flesh. Fiercely bright and extremely focused, at times it’s hard to disengage the actor from his Sherlock character. But as anyone who saw his Ice Bucket Challenge video will be aware (if not, check it out on YouTube), Cumberbatch is not a man who takes himself too seriously.

Dropping his guard a little, he continues on the subject of work-life balance. “I am really good at switching off and I can prioritise – and that’s really the secret: just knowing how to not get snowblind and focus on what’s right in front of you and when you’re not working, not to worry about the other things that you could be working on but just switch off.”

Unwinding after a hard day’s work is easier said than done, but Cumberbatch has found the solution: “You just close the door on the world – physically, literally and imaginatively, as well. You get used to your own company and focus on something very calming: something that’s not about you giving and the energy going outwards, but about energy coming back to you.

Like what? “It can be anything. It can be meditation, it can be watching a film, having a drink, just, you know, as long as it doesn’t involve dependency you can kind of choose whatever floats your boat,” he says.

Cumberbatch won’t have much time to switch off this evening. In a couple of hours those screaming fans will be waiting outside his premiere. For someone relatively new to the spotlight, Cumberbatch doesn’t seem flustered by the circus that surrounds him.

Did it take long to adjust to that level of fame? He pauses for a moment and then gives out a big laugh, “Erm… er… well the fact that I’m thinking about it means I still haven’t adjusted. It’s a constant process of negotiation and understanding and, yeah, it’s very odd but fun and I think I’m alright with it. It’s a learning curve; it’s a constant thing.”

And does he still get starstruck? “I still get completely starstruck,” admits Cumberbatch, who picked up an Emmy Award in August for his role in Sherlock. “I mean, really starstruck. I bumped into Aaron Paul and Bryan Cranston [stars of Breaking Bad]. And I had a complete meltdown, you know. Love them.”

Cumberbatch also stars soon in the final Hobbit film, The Hobbit: The Battle of the Five Armies, having appeared in the previous two instalments as the voice of Necromancer and the dragon Smaug in motion-capture form.

Even playing a wolf in a film about penguins, you can’t put your voice in a jar and send it to a studio

Cumberbatch is no stranger to the look and feel of a big-budget CGI blockbuster, having also appeared in Star Trek Into Darkness, but admits when he first saw himself as Smaug it was quite a moment.

“I actually sat there with a friend of mine who I’d known since we met at boarding school when we were seven years old. We both had the Hobbit books read to us by our fathers. Anyway, he and I were sat there in the cinema watching it [and I was] going, ‘Oh my God, I’m playing a fucking dragon’,” he laughs. “It’s unlike watching yourself.”

The physicality of a character is everything for Cumberbatch, whether he’s playing a dragon, an accomplished athlete – which Turing was, alongside having a great mind – or voicing an animation movie such as the upcoming Penguins of Madagascar.

“Everything has to come from the body,” he says. “You can’t, sadly, even playing a wolf in a film about penguins, put your voice in a jar and send it in a taxi to a studio and go and fuck off and have a holiday. You’ve got to be there physically, fully – and I love that.”

It was in the Square Mile that Cumberbatch filmed those memorable Sherlock scenes where he hurled himself from the top of St Bart’s hospital to perform his character’s ‘fatal’ leap at the end of series two. He will be returning to the City next year to throw body and mind into the most famous theatre role of all time. Tickets to see him play Hamlet at the Barbican sold out in hours, beating the demand for Beyoncé and Jay-Z’s On The Run world tour, which went on sale at the same time. “I’m a bit embarrassed about that,” he laughs. “But at the same time the fact that Hamlet has sold as fast as the king and queen of popular music on tour is just so thrilling,” adds the star, who made his Shakespeare debut at the age of 12 in A Midsummer Night’s Dream.

No doubt Cumberbatch has set aside a couple of front-row seats for his parents. Timothy Carlton and Wanda Ventham are also both actors and appeared alongside their only child in an episode of Sherlock.

But before Cumberbatch takes on the Prince of Denmark there is the small matter of next year’s Oscars. So how does he feel about being tipped for an Academy Award for his performance in The Imitation Game, which soon after this chat goes on to win the top prize at the Toronto Film Festival?

“It would be amazing but to be honest right now it’s so premature, it would be almost futile to talk about it. But I understand why it happens; it always happens at festivals when I’ve just had my suit from last year’s Oscars back from the dry cleaner,” he laughs.

“I think the most important thing is that any kind of buzz creates an interest in the film, which means people will see it, hopefully, and that means Alan Turing’s story will get to a broader audience. As a storyteller that is all I’m concerned about – and if I’ve done a good job playing him as well that’s great.”