Luke Treadaway is the National Theatre's million-dollar man. Now 28, in the last six years he has taken the lead in two of the most successful shows in the theatre's history. In 2007, he was Albert in the original production of War Horse, a show that has since transferred to the West End and Broadway, been seen by over four million people worldwide and inspired a Hollywood movie.

Last year, he was cast as Christopher in The Curious Incident of the Dog in the Night-Time, a production which also blazed into the West End, where it has already been seen by 150,000 people, and earlier this year won a record seven Olivier Awards – including one for him as Best Actor.

"It's a hell of a souvenir for the last 14 months," he says coolly of the trophy. "By the time it got to me on the night, we'd won five out of five, so I was completely prepared for mine to be the first one not to win. I was shocked. Then I went backstage and had my picture taken with [Best Actress] Helen Mirren. So, yeah. Fun night."

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It all began with a failed audition. In 2007, fresh out of Lamda, Treadaway tried out for a new play, The Five Wives of Maurice Pinder, at the National. He got down to the last two to play the teenage son, but lost out (to Adam Gillen). A couple of weeks later, the theatre's casting director called to ask if he might be interested in doing a workshop for a few weeks instead. "I remember my agent saying, 'It's called War Horse' and I wrote it down and went out to buy the book. I read it and thought, 'Wow. Oh my god, Albert's going to be an amazing part for someone. Can't believe I'm from Devon, too, but I won't get it. Never mind...'"

By the end of three weeks in a rehearsal room in south London – "It was before the puppets had even arrived from South Africa, so people were walking around whinnying, using an arm and a shoulder for a horse's head. It was a beautiful thing" – the part of Albert was his. And a fruitful partnership with the director, Marianne Elliott, had begun. Straight after the workshop, Elliott cast him in bit parts in Saint Joan. By night, he stood around behind Anne-Marie Duff, getting used to being on stage at the National. By day, he and Elliott wrestled with War Horse. When it came to The Curious Incident, Treadaway was first in line for the workshop – and once again won the lead. "I've never done a play with Marianne having auditioned for it. I think I'm pretty rubbish at auditions, really. I should just workshop everywhere."

The pair will reunite for the theatre's 50th anniversary gala in November. He can't say too much about it except that it will feature "scenes from big National shows over the years." After that, he is keen to work with his mentor on a third mega-hit. "She can have me whenever she wants me, put it that way. Whenever we see each other we always boringly talk about what we're going to do next. There will be something, I'm sure." Having played two teenagers, might it be time to stop waxing his chest and start playing… "Older than 15? Well, exactly. I played 14 in War Horse, 15 in Curious. I'm looking for a 16-year-old part next. How different would that be – playing an actual mid-20s man in something?"

For now, Treadaway is taking a well-earned rest from the theatre. He finished his West End run in The Curious Incident at the end of August, and after an all-night wrap party, hopped on a 5am plane to Spain. Today, fresh from holiday, he looks relaxed, dressed in a vest and jeans, grey Reebok hi-tops and a gap-year necklace in the shape of a fish. He is on a break – but it is the nicest kind of actor's break. (He's known the not-so-nice kind, too. "There have been hard times and great times and times in between," he says. "The odd six months out of work. Horrible.")

Hollywood has, unsurprisingly, come calling – and in the next few weeks he will head to LA to prepare for a big job his agent forbids him to talk about. Until then he is enjoying playing guitar in the Highgate flat he shares with his actor twin, Harry, actress girlfriend, Ruta Gedmintas, and musician friend Johnny. "I'd been thinking that when I finish this play, I must sit down and bloody play some guitar again. I love it, I really do. You have the time to decompress and realise that it's not all about acting. Because it's not."

When he was in The Curious Incident, he would struggle to shake off his intense alter-ego, talking in the teen's oh-so-deliberate manner for hours after a show. "When I'm in a job, I don't very successfully get away from it. I'm always ticking away. But when I'm not working, I have a similar commitment to seeing my friends and enjoying myself." Today, he is lightly hungover, craving a cigarette and given to lyrical outbursts, particularly about the part he has just left behind. "Will Christopher ever really leave me? I don't know. I think he'll always be a part of me or I will always be a part of him. Or something. I don't feel I've said goodbye to him forever. He's still there."

That said, he admits that it is a relief not to be going on stage tonight. Playing the autistic teenager, he spent a "very sweaty" two-and-a-half hours, five nights a week scrabbling on the floor, flinging himself into the air and running up walls. It took its toll – in the form of RSI in his arm and housemaid's knee. There is a new Christopher now and the entire cast has been replaced. "I think it was time. We all knew it and felt it. It's 14 months since we started at the Cottesloe and by the amount of elbow supports, painkillers and physio that everyone was having, it was clear that we'd all pretty much gone as far as we could go with it. For now, anyway." Which suggests a Broadway run could be on the cards.

In the meantime, he has other things to keep him busy. He is the star of The Rise, an offbeat British heist with Timothy Spall and Vanessa Kirby, which is released this weekend and is unusual for being set in Leeds, not London. "There are so many set in London, all 'fackin' cockneys'. It feels fresh," says Treadaway, drawing a veil over his own foray into the genre, the forgettable Get Lucky. Before that, his biggest film was the excellent Attack the Block. While he would love to work with Danny Boyle (he'd be an ideal heir to Ewan McGregor), theatre is his first love. "It's when I grow and learn most. You don't get time for that on a film set." He wasn't keen on Spielberg's War Horse, for example. "I'm not a very fair person to judge it, I don't think. First, I come from Devon and in my head I'd set it in my village. So they basically got it wrong. It didn't look like any Devon I knew."

The Devon Treadaway knows is, specifically, Sandford, where he was raised by his architect father and primary-school teacher mother. He has an older brother, Sam, who works in Bristol as a multimedia artist, and the aforementioned twin, Harry, another British rising star with roles in Control and Fish Tank to his name. Luke made his stage debut aged 3, playing a daffodil to his father's Big Bad Wolf in a local am-dram Little Red Riding Hood. From then on it was "non-stop" playacting with his ready-made co-star, Harry. "There was never a time you couldn't put a play on, because there was always someone to do it with," he says. "I'd give Harry the odd cameo."

Growing up, the twins worked to carve out separate characters, never going into school with new shoes or a new haircut on the same day, never wearing the same clothes. "I think that's a bit strange to do that to twins. They've enough of an uphill struggle to make their own identity. It's weird." But they also loved all of the same things. They both played on the same rugby team. They both played in the same band, Lizardsun. Aged 16 they both joined the National Youth Theatre, where they both instantly befriended Matt Smith, aka Doctor Who. "He was leaning out of the window and was the first person we met when we arrived. We've been best mates ever since." Aged 18, they both enrolled at Lamda.

"We weren't going to draw straws or toss a coin to decide who couldn't go. That would be stupid." And aged 19, they landed their first professional job, a film called Brothers of the Head, in which they played conjoined twins, and spent 15 hours a day stitched into a wetsuit together. They have worked together since, in Mark Ravenhill's Over There, a play about East and West Germany, which climaxed with Luke, wearing nothing but high heels and a smear of ketchup, kissing Harry.

It must get a little heated in their flat – four performers, all scrapping for jobs. "We're all ambitious and and work with good people and do good scripts, so it's competitive in that sense. There's a healthy desire to get good jobs. But I don't hide Harry's script to stop him going in for an audition, or lock him in the bathroom so he can't go for a meeting." Would he do another job with his twin now? "They do come our way. I don't regret doing it. If it had ended up that no-one wanted to cast us on our own and we'd just ended up playing twins, then it would have been a really bad decision. But we've both done our own thing. It's worked out."

'The Rise' is in cinemas from 20 September and on DVD from 30 September