Class act: Is Olivia Colman Britain's most versatile actress?

Olivia Colman has the fear – the one actors get when they don't have another job lined up. "Maybe I've had a good run of it and now it's all going to stop," she says. Right now such a dismal outcome is highly unlikely, as Colman seems to be everywhere – as Queen Elizabeth (later the Queen Mother) visiting Bill Murray's Franklin D Roosevelt in Hyde Park on Hudson, as a foul-mouthed marriage counsellor in the British rom-com I Give It a Year, and as Nick Frost's sister in the eagerly anticipated dance comedy Cuban Fury.

Also in the wings are a guest turn in the wonderful Sky Atlantic comedy This Is Jinsy and a new series of BBC2's dog-collar sitcom Rev, while for the next eight weeks she is going to be sharing top-billing with David Tennant in the ITV whodunit Broadchurch – a sort of British The Killing about the murder of a boy from a close-knit Dorset seaside town.

"The Killing was so brilliant that it seems a bit up yourself to say this is 'the British Killing'," says Colman, characteristically averse to hyperbole. She plays a local police detective miffed when an abrasive outsider (David Tennant, giving good stubble) is parachuted in to lead the investigation, while an all-star cast of suspects (Andrew Buchan, Jodie Whittaker, Vicky McClure and Doctor Who's Arthur Darvill) slowly reveal theirf secrets. Colman's is a very personable breed of copper (Scott & Bailey are a pair of Dirty Harriets in comparison), which is hardly surprising since she's modelled on herself. "I wanted it to be me in the police force," she says. "I thought I could damage that by doing too much research, which is my lazy way of saying 'I couldn't be bothered'."

Take that with enough salt to grit Heathrow airport, for the rise and rise of Olivia Colman is about unassuming hard work as well as talent, intelligence and (never a quality to be underestimated in her profession) likeability. Her success is certainly nothing to do with being brazen or pushy – she positively blushed with self-consciousness on our two, albeit brief, prior meetings (on the sets of Broadchurch and This Is Jinsy). Today however she's more relaxed, perhaps because it's her 39th birthday, and also maybe because the ice had been broken with a discussion about the attractions of her native north Norfolk.

There's no East Anglian accent; in fact her voice reminded me of a softer version of Rising Damp's Frances de la Tour – Miss Jones's gentler, more diffident younger sister perhaps. And like another shy comedy actress, Hunderby's Julia Davis, she has a dazzling smile that seems designed to deflect the inane intrusions of journalists and other idiots. "Perish the thought," she says when I ask whether she has a Hollywood agent, and she describes herself as "not terribly thick-skinned", when explaining why she doesn't tweet and why she keeps "my head down" in public. What do people say when they do recognise her? "They shout Peep Show," she says, mimicking a drunken cockney. "I'm not sure what I'm supposed to say in response."

In a way these boors are right – it was Peep Show that first made Colman. In Sam Bain and Jesse Armstrong's brilliant (and brilliantly sustained) Channel 4 sitcom, she played Sophie, the endlessly forgiving girlfriend of the appalling Mark (David Mitchell). "I couldn't do the latest series because I was doing Broadchurch," says Colman. "But also from a story point of view I don't think Sophie was as interesting for people to watch any more. And Dobby (played by Isy Suttie) is such a brilliant character and it's more fun to watch Mark fuck it up with other people."

Also, she might add, in recent years she has had to make the conscious decision to distance herself from her co-stars, David Mitchell and Robert Webb, whom she first met in Cambridge. "That was the discussion I had with my agent," says Colman. "She said, 'I know you love them and I know you're probably going to cry but you need to decide what you're going to do'. And Rob and David were heavenly about it. I'm so grateful for them – they were my first job and without them, well, I could be one of my many mates who doesn't get much work. But it's their show – it's Mitchell and Webb, not Mitchell and Webb and Colman. So Lindy [King – her agent] was right and it's all worked out."

It has indeed, starting with Tyrannosaur, the directorial debut of actor (and Colman's friend) Paddy Considine, with whom she had starred in Shane Meadows' mock documentary Le Donk & Scor-zay-zee, and with whom she has just finished filming an upcoming episode of the ITV period crime drama, The Suspicions of Mr Whicher. Colman's performance in Tyrannosaur, as a charity shop worker enduring hidden domestic violence and further challenged by Peter Mullan's raging alcoholic, was a revelation, and won several awards, including the Special Jury Prize at the Sundance Film Festival.

"It changed everything, really," she says. "It's funny… I've always done bits of drama, but clearly nothing anyone had ever seen." Next came her Carol Thatcher in Abi Morgan's misjudged Margaret Thatcher biopic, The Iron Lady, being singled out in Meryl Streep's Bafta acceptance speech for being "divinely gifted". Colman's research involved watching tapes of Carol in the 2005 series of I'm a Celebrity… Get Me Out of Here!. "It was such a godsend being able to watch the person you're trying to emulate in that situation," she says. "It was her being herself. Anyway, I thought she was amazing; if I was stuck in a forest I'd want her with me."

Colman was born in north Norfolk in 1974, the year before Margaret Thatcher became leader of the Conservative Party. She had an "absolutely heavenly" childhood moving around the county as her father, a surveyor, and mother, a nurse, renovated houses. "I had a lovely, feral, free childhood – out and then come back when you're hungry or it gets too dark," she says. "I feel slightly cruel that I'm not offering my children the same." The children are her two sons, Finn, aged seven, and Hal, five, with her husband, writer Ed Sinclair – but more of them later.

Colman didn't discover acting until she went as a sixth-former to the independent girls' school Gresham's in Holt, but the idea that you could make a living from this new-found passion never crossed her mind. "I'm the only weirdo in the family who's gone into it," she says. f "My older brother, who was in the Army, now owns his own building company. My half-sister was a nurse and is now a psychotherapist."

Instead she went to teacher-training college in Cambridge, but left after a year ("I was rubbish… a generation of children have had a very lucky break"), but not before that fateful meeting with Mitchell and Webb when she attended an audition for the Cambridge University Footlights under the mistaken impression that it was a straight drama society. "I'd never heard of Footlights," she says. "I think Rob and David probably thought I was quite game because they said, 'Find something in here and try and sell it to us', and I picked up a cigarette butt and was trying to sell it to them as nutritious and then ate it. Instead of laughing they just looked slightly shocked."

Another long-standing relationship that began at Footlights was with her future husband, Ed. "We did a play together," she recalls. "He did a play because he fancied the director and I did it because she was a friend of mine, and then I turned up and said (in a breathy voice now very much like Miss Jones in Rising Damp), 'Oh, he's amaaazing…'. And luckily the director… she wasn't that interested in him… thankfully… so, erm, then he got into drama school [the Bristol Old Vic] when he graduated and I was heart-broken that it would all end there. So I said I'd come along and I could support him."

A determined suitor, I suggest. "Yuh," agrees Colman, who thought that drama school looked like so much fun that she herself applied. She and Ed got married seven years later ("He's a steady chap… he likes to be sure"), although Ed has given up acting for writing. "I've got so many friends who just aren't working, it's puzzling; I've no idea why," says Colman. "And for Ed the work just stopped. But actually for him I think he was quite relieved because he always wanted to write and he's brilliant at it. I was lucky enough to pay the rent for both of us. It's not fair if I'm living the dream and he's not."

Wow. Would she consider marrying me? "Ha. Well, I'm hoping the novel will sell for a lot of money," she says. "I've only read the first four chapters. It would be awful if it came out and it was shit."

I ask whether her sons have seen her in anything, desperately racking my brain for any child-friendly fare that Colman might have appeared in. "I did Doctor Who, thinking there's something they can watch," she says. "I turned it on to be horrified and tried to turn it off again because there was mummy with a big scary mouth [in the episode, The Eleventh Hour, Colman's character sports piranha-like fangs]. I didn't really think it through."

But back to drama school in Bristol. After graduating, Colman set about looking for straight roles, but always seemed to gravitate towards comedy. It's at this point, with faultless comic timing, that her mobile phone goes off, her ringtone a honking clown's horn. Her screen-saver is a picture of her dog, a 'Jackapoo' (a Jack Russell/poodle cross) called Alf. "They're not supposed to moult," she says. "But I keep finding hairs."

There's something canine about Colman's nickname, Colly – she had to change her birth name, Sarah, to Olivia, because of an Equity clash with another Sarah Colman. "One of my best friends at university was called Olivia and I always loved her name," she says. "I was never Sarah; I was always called by my nickname, Colly, so it didn't seem so awful not to be called Sarah."

Colman's knack for comedy saw her progress through such shows as People Like Us, The Office, Black Books, Green Wing, as well as the aforementioned Mitchell and Webb collaborations. But it's only with her two most recent sitcoms, Rev, in which plays the wife of Tom Hollander's inner-city vicar, and Twenty Twelve, the Olympics comedy in which her PA shyly won the heart of boss Hugh Bonneville, that she has parts large enough to equal Peep Show's magnificence. And Rev has won her a whole new fan-base.

"We went to try and do some filming at Greenbelt Christian rock festival," she says, "but we couldn't use the footage because everybody went 'Aaah!'… It was like being in the Stones. There was a vicar shaking Tom's hand and saying, 'Thank you so much… I'm proud of my dog collar – you've shown us as fallible and human and trying our best'."

Rev returns at the end of the year, although Bad Sugar, the telenovela pastiche co-starring Julia Davis and Sharon Horgan, surprisingly won't be progressing beyond the pilot stage that was shown on Channel 4 last year. "It's been decommissioned," she says. " I think there might be some new blood and it's not their baby… I don't know… politics."

And then our time is up and it's off to have her photograph taken – and I notice that this seemingly self-effacing actor, casually attired in black leggings and Ugg boots, is carrying a plastic bag with a pair of ultra-high heels, in the fashionable dominatrix style, and a birthday present to herself. It leaves me with a nagging feeling that there might be another Olivia Colman that I haven't yet begun to meet.

'Broadchurch' begins on ITV this Monday