WHAT to do on meeting a double Oscar-winning actor playing a triple election-winning prime minister?

It's the sort of matter etiquette guides don't cover, so Olivia Colman went with her instincts and gave Meryl Streep a playful biff on the arm.

"I didn't know what to do with myself," says Colman, who plays daughter Carol to Streep's Margaret Thatcher in The Iron Lady. "She looked like Margaret Thatcher, which is pretty intimidating and scary anyway. And I knew it was Meryl Streep under it, which is also pretty intimidating."

This is said in such a breathless, oh-gosh way you might never guess the speaker has just won the British Independent Film Award for best actress (for Tyrannosaur), is a cult comedy heroine courtesy of Peep Show and Rev, and seems to be on more "people to watch in 2012" lists than Hillary Clinton and Beyonce combined. Colman, refreshingly, is not your flint-faced diva type. Naturally wide-eyed and dressed in a fake fur coat, she looks like a woodland creature who has wandered indoors to a PR firm's Soho offices for a heat rather than a rising star punting a picture.

Heat has already come The Iron Lady's way, with former ministers and aides queuing up to condemn Phyllida Lloyd's biopic, usually on the basis of the trailer alone. One called the film indecent and exploitative for showing Thatcher as an infirm old lady.

Although "on the other side of the fence, views-wise" from Thatcher, the film made Colman, 37, appreciate what a mould-breaker the Tory leader had been. "It is extraordinary that a woman fought against a sea of men, and a sea of men who didn't know how to cope with women anyway because they were all public schoolboys. To rise to the very top of all of them, amazing. Everything else is a matter for debate." She pauses and laughs. "I'm trying to be diplomatic about it."

What has irked some Thatcher supporters is the way the film highlights her current frailty. Unfair? "It would be unfair if we were laughing at her predicament. That would be appalling." Instead, says Colman, the film shows in a tender way how even the most powerful person can be made fragile by old age. "Sickness doesn't care who it is attacking."

Whatever the reaction to the film in 2012, last year was a very good one for Colman, with critics hailing her portrayal of a battered wife in Paddy Considine's demanding drama, Tyrannosaur. Also starring Scotland's Peter Mullan and Eddie Marsan, it looked like a tough gig.

"Actually it was one of the most enjoyable shoots I've ever had, which sounds weird having seen the film. We all felt we were part of something very special. Those are the sort of films most actors want to do. I can't imagine anything topping that experience, ever."

And there was always Mullan to cheer things up. "He's such a raconteur, he's got a million stories and he'll have everyone in stitches then go straight into a scene where he's ...." At this point three things happen. She segues into a Scottish accent, suddenly looks horrified, and starts to apologise. "Sorry, I can't believe I did that accent to you. How awkward." I'm too busy laughing at the impersonation to be offended.

To those who only knew Norfolk-born Colman from Peep Show, Tyrannosaur would have been a surprise. She first met David Mitchell and Robert Webb in Cambridge. They were "clever, clever" students at the university, she was doing a teacher training course. She doesn't quite know why they clicked. "They're two of the loveliest people I know and they always were. They haven't changed. We were just a bunch of 20-year-old idiots. Slightly bumbly. We just got on."

Colman didn't become a primary school teacher – "There's a whole generation who should be grateful for that" – and instead went to drama school in Bristol.

Having appeared in Hot Fuzz, Green Wing, and That Mitchell and Webb Look, she achieved spotted-in-the-street recognition as Sophie, David Mitchell's demanding girlfriend in Peep Show.

What sort of a girlfriend was Colman? "It depends who the boyfriend was. I think I was quite a good girlfriend. I never really went in for drama.

"I used to watch some of my lovely, witty girlfriends being dramatic with boyfriends and think that was a bit boring and tiring. I was quite fun. Probably more fun than Sophie."

Sophie certainly gives heightened meaning to the word intense. Particularly during those childbirth scenes, which turned out to be remarkably similar, in sound effects, to Colman's own experiences. "Apparently I mooed," says the mother of two. "My husband said it sounded quite like a cow. I said that's nice, thank you."

At the prompting of her agent she decided to start branching out on her own. "I was so terrified about telling [Mitchell and Webb] because I love them so much. They were so supportive, with Rob saying 'We knew it was only a matter of time'."

It was a gamble, she admits, but it is paying off. Besides Tyrannosaur and The Iron Lady, she's on stage in London in February in Noel Coward's Hayfever, and she has just done a pilot for a new Channel 4 comedy series, Bad Sugar, a British take on Latin American telenovelas. Also starring Nighty Night's Julia Davis and Pulling's Sharon Horgan, written by Sam Bain and Jesse Armstrong (Peep Show), and directed by Ben "The Inbetweeners" Palmer, the omens for Bad Sugar are good.

Colman, Davis and Horgan came up with the original idea, though she's keen to give the credit to her co-stars and particularly the writers. "I couldn't have written it and I wouldn't have the confidence to think of an idea without them." Really? "Definitely. I haven't done it up till now. Lots of people think of things all the time. I never did. It's clearly not what I naturally do."

It's hard to reconcile this view of herself with the opinion others, Mitchell, Webb, Considine and the rest, obviously have of her. She, however, is more comfortable talking up their talents rather than her own.

"It's very nice that people might think it's a collaboration but that would be wrong of me to take that glory. We all get on very well. Maybe there's something about me that makes them think of something. It would have happened without me."

Her children won't see her in The Iron Lady – not really the thing for a four-year-old and a six-year-old – and they've been too young to catch her other stuff (the Doctor Who part was far too scary). But many others will see her soon as Carol Thatcher, false nose and all. She was glad of the prop.

"It made me feel a bit more confident. It's scary playing someone we all know so well. I'm slightly terrified about her seeing it and saying 'it's nothing like me'. It's a piece of fiction with a hint at her. I can't do impressions so I'm hoping people will forgive me. And they've got Meryl to watch."

The Iron Lady opens in cinemas on January 6.