Olivia Colman is insufferable. We've been sitting for an hour on the balcony of the Ritzy cinema in south London and she's given me nothing but unremitting cheeriness. Doesn't she realise I need dirt, self-disgust and something really vile about working with Rose Byrne on I Give it a Year and/or Bill Murray on Hyde Park on the Hudson? She's even managed to be positive about the view. "Look at those gorgeous trees," she says of the espalliered Parisian-style Brixton avenue below. "How do they make those trees square?" Vigorous and regular pruning, no doubt, I reply glumly. "Lovely aren't they, though," she says, with that sunny smile that bewitched viewers when she won two Baftas in May.

She's been unacceptably sweet about everyone she's ever worked with. Paddy Considine, who directed her as a posh charity shop worker who kills her abusive husband in the 2011 film Tyrannosaur? "He's one of the most beautiful humans I think I've ever met. He's utterly good. His family and his wife, too." David Tennant, with whom she starred in ITV's recent hit cop drama Broadchurch? "An angel and absolute sweetheart." Tom Hardy, with whom she will soon star in the film Locke? "He's proper stuff of legend." Katie Leung, one of her co-stars on Channel 4's new drama series Run? "Oh my God, she's amazing." What was it Bill Murray said sarcastically to Andie MacDowell when she was similarly chipper in Groundhog Day? "Gosh, you're an upbeat lady."

And then there's Meryl Streep. It was Streep who called Olivia Colman "divinely gifted" during her 2012 Bafta acceptance speech for her performance as Margaret Thatcher in The Iron Lady (Colman played her daughter, Carol). Reminded of this, Colman squirms eloquently. If there were a Bafta for Best Performance of I'm-not-worthiness in an Interview Scenario (and there really should be, though the winner's speech would be predictable), Colman would now have three on her mantelpiece. "That she even remembered my name is exciting. She's like – Meryl Streep!"

She admits to rewinding and replaying the Streep encomium. Perhaps, after all, this suggests the dark, obsessive side to Colman's Pollyanna-ish personality the journalist needs, instead of all this useless positivity. I imagine her settling down for hours with the remote. Divinely gifted. Pause, rewind, play. Divinely gifted. Pause, rewind, play. All the time watching the screen with that slightly demented expression – mouth agape, teeth bared, eyes glazed – that she gave her PA character Sally Owen as she forced oversized pastries on Hugh Bonneville in lieu of sublimated passion in the Olympics sitcom Twenty Twelve. It probably wasn't like that.



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"People said: 'What's it like with Meryl Streep?' and I wanted to make up some shit, but she's lovely. She's a mummy and she loves her craft. She loves what she does. There's no vanity, no ego, she's a really nice woman. I think they're the best actors."

The Streep model is the key to understanding Olivia Colman. Versatility plus public niceness multiplied by a secure private life equals professional success. "Eddie Marsan [superb as the chilling coward of a wife beater in Tyrannosaur] said the ideal is to have an extraordinary career and an ordinary life. That's so right. My priority is my family. And if that's all OK you can branch out and hopefully do good work."

During the previous decade, Colman wasn't known for Streep-like versatility, but as foil to poncy Cambridge graduates David Mitchell and Robert Webb during the first seven series of Channel 4's cult sitcom Peep Show. She was Sophie, the posh muppet who succumbed insanely to the non-charms of both Mark and Jez before dallying with her no less tragic but more butch workmate Jeff. Colman held her own with the often coarse material there and in That Mitchell and Webb Look (there was a particularly memorable scene in which they reflected on the benefits of home working from the point of view of maximising opportunities for self-abuse). In The Green Wing and Rev, too, she made us laugh by playing women suffering from the inanity of men.

Olivia Colman with David Mitchell in Peep Show … suffering from the inanity of men. Photograph: Channel 4

So we knew that Colman could do funny, but few would have then imagined, least of all Colman herself, that by 2013 she would be written up as an actor of Streepian versatility, nor that she was capable of incarnating so convincingly a string of downtrodden women in some of the most shattering roles in recent British TV and film drama. But that's what has happened. "Olivia Colman is to acting what Germany is to car making and gravity is to the universe – she is technically excellent and manages to be everywhere all at once," wrote Stephen Armstrong in the Radio Times after she won best supporting actress in Jimmy McGovern's Accused and best comedy performance for Twenty Twelve. "You could even say she is to acting what Gareth Bale is to football: hailed by peers, critics and millions of viewers." When will this festival of niceness stop? Not soon. The Daily Mail's eulogy to her suggested that she is becoming the new Helen Mirren, which was intended as a compliment.

She's finding the adulation embarrassing. "It's slightly scary, that tall poppy syndrome. It could all go wrong. I don't know. It's weird," she says and for a few seconds that toothy grin disappears. "It's weird."

Worse than weird. Awards and media love-in have a downside. "After the Baftas we were followed by a car, which I found really upsetting." She means the press snappers were on her case. "I'm a mum eating a sandwich with my kids. How is that going to sell newspapers? She has a point: in this post-Leveson media milieu, why should Colman, a member of the anti-press intrusion group Hacked Off, have to put up with that? But to play devil's advocate, it might sell newspapers. Who wouldn't want to know what Olivia Colman has in her sandwiches?

Olivia Colman at the Baftas. Photograph: Mike Marsland/WireImage

We're meeting because Colman is starring later this month in Run, a four-part drama set on a south London council estate – a stone's throw from where we're sitting. She plays disempowered matriarch Carol – as hard as nails, like EastEnders' Lou Beale, but as brittle as pressed flowers when confronted with the horrible truth that her teenage sons have turned out bad 'uns.

Her performance reminds me of Lesley Manville's as a similarly downtrodden mum from a bleak south London estate in Mike Leigh's 2002 film All or Nothing. And no wonder: Run's writers Daniel Fajemisin-Duncan and Marlon Smith cite Leigh along with Spike Lee as inspirations for their work. In one key scene in Leigh's film the overweight teenage son (James Corden) has a heart attack on the estate, and Manville – pinched, mousy and throughout poised to cry over her lousy lot, cradles her massive boy where he lies. "Ooh that sounds right up my street," says Colman when I tell her the plot.

Run is equally unremitting in its bleakness. It even includes an homage to the dismal karaoke scene in All or Nothing, in which Colman and her mate do a wretchedly toneless retread of Cyndi Lauper's True Colours. If there is a Bafta for worst vocal performance (and there really shouldn't be), Colman's a shoo-in. "Thanks a lot!" says Colman. "I was really proud of that performance. I actually have to say I tried my hardest." She's got to be kidding. "You could have broken it to me gently. That's the musical career ended then." There's nothing like being teased by actors.

"They were doubtful in the audition if I would be able to do the part," says Colman. Why? "Well, because you turn up and go 'Hello!'" Colman simulates a snooty voice, or rather a voice snootier than her already genteel tones. So the director thought she was a Cambridge posho? "Yeah, they did. But actually I was at the teacher training college, not the university proper."

I can't help laughing at that very British "actually", that fastidious demurral over status. Cambridge was where she met Mitchell and Webb, but she wasn't part of the elite like them, even though she had been to the posh Gresham's boarding school in Norfolk (alumni: Britten, Auden, Sir James Dyson, the woman born Sarah Caroline Olivia Colman in 1974) before going to Homerton teacher training college in Cambridge. It was acting with the likes of Mitchell and Webb there that induced her to give up her vocation as a primary school teacher.

"They were bloody clever, but I wasn't and I left after a year," she says. Colman does this a lot – modestly playing dim when she isn't. It's a compelling performance, the national treasure reviewing her shortcomings. "So it feels a bit bad when people assume I worked as hard as they did, because I didn't." Oh come on, they probably didn't work hard. "No I don't think they did – but they probably didn't need to because they're so clever."

Back to grim south London. At one point, Colman's long-suffering mum makes a Jamie Oliver dal for her boys. "She will have seen stuff on telly with perfect families and thought, I can do that." But the boys complain that the curry is green and leave to get some fast food rubbish. Graceless monsters. Nobody walks out on Olivia Colman.

Colman finds this family dynamic almost unbearable and starts to well up at the memory of the framed family snap they used on set depicting the teen horrors as little poppets. Why does she find that so affecting? "It was a golden time and they loved you and that changes and they don't want to talk to you any more and they bugger off. Heartbreak. Awful." Are you all right, Olivia, I say as she wells up. "Bit wobbly, I'll lean forward. I can't bear the thought of my kids turning out like that."

Colman has two sons, Finn, seven, and Hal, five, with her writer husband Ed Sinclair, whom she met at Cambridge. "I can see why people keep having babies. We were looking at a school for my youngest this morning and there were all these little boys and girls. So sweet. And then the teenagers walk past and, my God, they're enormous and I bet they don't kiss their mummies. I'm just going to force my children to remain lovely."

Olivia Colman on stage in Hay Fever … 'I made really bad mistakes and got the giggles.' Photograph: Tristram Kenton

Good luck with that. Colman has previous in parlaying her teary self into dramatic poignancy. When she played DS Ellie Miller in Broadchurch, she cried reading the script about the murder of 11-year-old Danny Latimer. "It's just awful, the idea that your children could go before you," she says. While David Tennant was an out-of-town detective drafted in to investigate the killing, she was the local cop, overwhelmed by the murder of a boy she knew.

Colman recalls visiting a mortuary while working on Broadchurch. "The man who ran it was just beautiful and respectful. I thought if I lost anyone I loved I wouldn't mind entrusting them to this beautiful person. And then this beautiful boy Oskar [McNamara] who played Danny – we weren't shown him before we filmed. We just came in and there was this child lying dead. Even though the script says you're hardened, I just couldn't bear it and started sobbing. It was awful.

"Since having kids, I find things much rawer. My priority is my family and I can't bear to leave them." One of the lures of taking the role in Run was that filming was near her south London home. She nearly turned down the role in Broadchurch after realising filming would take place in Somerset and Dorset and she would be away from her kids. It's almost inconceivable that she would go somewhere as distant Hollywood at this – what was it she called this period of family life? – "golden time".

But there's a problem with her career trajectory. Colman is yet again playing a downtrodden woman. She's been a victim of domestic violence in Tyrannosaur, a woman taking a stand against gang culture with Anne-Marie Duff on their grim estate in Accused and now, in Run, she's a woman whose life is composed of petty thefts, getting thumped by her ex and being scorned by her sons. She does worry about getting typecast. "After Tyrannosaur came out I got five or six scripts about women who were victims of domestic violence who take revenge on their husbands. I thought, 'people are going to know the ending of this'."

She denies being typecast. "In Hyde Park on the Hudson I was the Queen [ie the wife of George VI]. Hardly downtrodden." Was it tricky to impersonate a real person? "I think I got away with it because nobody remembers what she sounded like and anyway, everybody was watching Bill Murray." What about when she played Carol Thatcher in The Iron Lady? "That was more difficult because everybody knew what she sounds like. I watched her on I'm a Celebrity to get her voice right."

She never met the late prime minister's daughter, but warmed to her nonetheless. "If I was stuck in the jungle I'd want her on my side. I imagine we wouldn't agree on a lot of things, but I liked her and the nation did." To be fair, most of the nation didn't watch Carol Thatcher on I'm a Celebrity.

What next for Colman? We will see her in a second series of Broadchurch. But surely the storyline was wrapped up at the end of the first series? "I know what the premise is for the second one but I don't know if I should tell you."

Her diary is relatively free then for her to fulfil her manifest destiny as the first woman Doctor Who. "My brother texted me yesterday and said: 'Congratulations – you're 14-1 at the bookies for Doctor Who.'" If Colman truly is a national treasure, the Gareth Bale of acting and the new Helen Mirren, then surely Matt Smith must regenerate into her later this year. Worth a bet? She shakes her head. "I imagine they've already approached the people they're thinking about."

If not the Doctor, what about the first female 007? "Then you'd have to be really energetic, wouldn't you? I couldn't compete with Daniel Craig coming out of the sea." If you're imagining Colman coming out the sea half naked now, stop such treasonous thoughts immediately. She hasn't done topless since she and Robert Webb played naturists in the ill-advised 2006 film Confetti.

But perhaps she wouldn't be good in either role, because she can't act. This, at least, is Hugh Bonneville's theory: "Olivia Colman can't act. There, I've said it. She really can't." Fantastic stuff: if only he'd have stopped there, we might have been able to really get the Olivia Colman backlash going. That might stop her being so intolerably cheerful. But he didn't, damn him.

"She can't act because she can only be: she has a phenomenal ability to be utterly spontaneous in every role she plays. Her comedic and dramatic range is extraordinary, as is her natural gift of being loved by everyone she works with. What a cow."

She laughs as I quote this. "He's the same, I think. I love it that he said it, but a lot of actors are like that. I suppose that as you get more confident and better scripts it's easier to commit to it and be more truthful and imagine how that person would feel." In other words the appearance of a performance disappears and only naturalness remains.

She can do that on film and TV, and perhaps even in interviews. But, she says, she can no longer achieve that actorly alchemy on stage, even though she trained at Bristol Old Vic after Cambridge. She winces when I remind her of her last stage performance in Coward's Hay Fever last year. "I don't think I did a very good job of it." The critics, though, were hardly damning: while Michael Billington reckoned "Colman does no more than she has to as a predatory vamp", Kate Kellaway thought her "outspoken Myra is impeccably judged".

But the critics missed the worst, she says. "It was later in the run that I made really bad mistakes and got the giggles. West End audiences haven't paid to see that. I felt really bad about that. Oh dear."

When she goes to the theatre, which she does a lot, she says, it deepens her sense of inadequacy. What a masochist. "I find Shakespeare terrifying. When Simon Russell Beale does a speech I understand every word of it, but if I did the same speech people would be going 'Huh? What?'" Nonsense: she'd be a terrific Lady Macbeth, ideally channelling the borderline deranged Sally Owen. Imagine her snarling through those teeth at Bonneville's pathetic Macbeth: "Give me the daggers. The sleeping and the dead/Are but as pictures; 'tis the eye of childhood/That fears a painted devil." Superb. We need to see more of Olivia Colman's dark side.

• Run starts on Channel 4 on Monday 15 July