Olivia Colman first made her name in comedies such as Peep Show and Twenty Twelve, but in the last year she has exploded into drama. Megan Conner talked to her just as Britain was holding its breath for her performance in the finale of Broadchurch

Olivia Colman has asked to meet at a private members' club in Soho. I didn't have her down for that – she doesn't seem the type to be falling out of the Groucho, but this one is a little less hip, the sort of place that is filled with film and television executives having nice lunches. On the table next to us a group of agents is discussing whether the Cannes film festival is up to much this year: "I mean, I might be better off in San Sebastián…" I think I hear one sigh.

It's the sort of place you might come if you were a bit of a luvvie or were looking to get in with the BBC. But it soon turns out that Colman has a more pragmatic reason for being here. Today is not a normal day in the life of the actor. Today brings the finale of Broadchurch, the ITV drama which has turned her into a household name. Tonight nearly 10 million viewers will tune in to find out who killed schoolboy Danny Latimer. "I'll do anything to avoid being asked who it is right now," she laughs, pouring the tea.

She arrived by black cab, which is not her usual style, but: "I've been spending a fortune on taxis this week. It got to the point where I had to stop getting the train because people kept asking me, and well, I was afraid of letting it slip. Obviously," she adds, "there's been no chance of me going out and getting pissed. I'd fuck it up instantly. People would only have to look at me and I would tell them who it is."

Colman dispels any remaining possibility of luvviedom by carefully assembling me a forkful of her lunch because she insists I must try it. We are both eating mackerel – mine has been grilled, while she has ordered the "soused" option, mainly because it "sounds fun".

"I'm getting you a good bit of everything," she says, gathering up the ingredients from her plate. "I want you to try this nice mustardy cream. Here we go. Do you mind sharing my fork? Sorry."

She keeps conversation light and is endearingly polite, more timid than I expected, even a little shy. She does not look worlds apart from the characters she plays on television, whether that be DS Miller in Broadchurch or Twenty Twelve's Sally, just better dressed; younger, even. I am intrigued when she says she is quite a mellow person ("I mean, I rarely shout and scream"), because on screen she has a habit of going ballistic and doing it brilliantly. In the final episode of Broadchurch the focus became less about the killer – who was revealed (spoiler alert!) as Miller's husband Joe – than Colman's reaction to the news. The impact of her performance was so raw and immediate her name started trending on Twitter – hardly unusual for the show, which has become TV's most tweeted-about drama (according to ITV), but a concept I know will be lost on Colman, having spent five minutes at lunch explaining to her what a hashtag is.

She claims not to be on Twitter because she is "not thick-skinned enough". Until recently her roles in TV comedies such as Peep Show only saw her vaguely recognised on the street: "I think people knew the face, if not much else, and so I always got a bit of pointing and shouting."

Cliff hanger: Colman with David Tennant in ITV’s surprise hit, Broadchurch. Photograph: ITV

But the last two years have proved a revelation. As well as being a great comedian, Colman has shown herself to be a breathtaking tragedian. The turning point was Tyrannosaur, an independent film directed by actor Paddy Considine in which she played an abused wife. Not only did she receive a clutch of industry awards, but casting directors finally realised what Colman was capable of. "It was weird," she admits, "because I'd been doing drama auditions for years."

Offers of grittier work streamed in: first the BBC thriller Exile, dealing with Alzheimer's, then the Margaret Thatcher biopic The Iron Lady, co-starring Meryl Streep. As Colman mastered the ex-PM's daughter Carol on the big screen (she says by watching old episodes of I'm a Celebrity…), Streep was said to be awestruck and declared Colman "divinely gifted" in her Bafta acceptance speech. Colman was asked to give a statement to the press on the day of Thatcher's death, an idea which still makes her guffaw: "I mean, I can't imagine what they thought I might be able to add."

This year she is up for her own Bafta. Well, two, actually – a nomination for both comedy and drama, recognising her diverse talent. One is for best comedy performance – her role as Hugh Bonneville's lovelorn PA in the brilliantly nuanced Olympics sitcom Twenty Twelve – while the other is a best supporting actress nod for Jimmy McGovern's Accused. She describes the latter, in which she played a mother grieving the shooting of her son on a Manchester council estate, as "beautiful. I mean, when people say there's nothing on British telly it's just nonsense." As in Broadchurch, the power in Colman's performance is weighted in the final scene, for which she says she was given the gift of a "wonderful speech". "Oh god…" she says, breaking off. "I'm getting a bit teary just thinking about it."

Unlike her method chums, Colman is someone with her emotions always close to the surface. "I've got a lot of actor friends who go: 'I'm in the process of working it all out'," she says, mimicking a panicked face. "And I suppose I just don't. I don't annotate scripts, I don't think too deeply about it, I just have to… feel it. Oh, it sounds awful, doesn't it? Like I don't deserve to be doing this."

She cannot explain much more: "It's just how I do it. I don't know if that's what you're meant to do; I don't know if it's what other people do. I just think you can only imagine how you would feel – if it were you…"

She knows instantly when a scene hasn't gone well – "I don't believe myself. I'm literally just saying the lines and I think that's bad and I can tell" – but at the end of a hard day's work she's not so bereft of energy she can't make it to the pub. "If it's a tough scene it's pretty cathartic, so I can generally just order a gin and go: 'Aah…'"

Recently she reunited with Paddy Considine. The pair will star in The Suspicions of Mr Whicher, returning to ITV next week. You sense that she is a little bit in love with him. "Oh I am, because he's just so black and white, can't lie, and once he takes you into his heart that's it." Considine has also bonded with Colman's husband Ed Sinclair, a writer: "Ed sent me this beautiful email after he saw Tyrannosaur, telling me what he thought, so I forwarded it to Paddy." And? She laughs. "Paddy rang me and went: 'He got it. He fucking got it.'"

Originally from Norfolk, Colman got the urge to act while starring in a school play at the age of 16. Unable to see how she could make it a career, she enrolled for a teacher training course at Cambridge, where she joined a golden generation of talent at Footlights. It was here that she met Sinclair, along with her future Peep Show co-stars David Mitchell and Robert Webb. "It was the first time I met people and thought: 'Oh, you're like me…'" she says. "You like the acting... Oh, I see." Yet she claims not to be Cambridge proper: "I was no Stephen Fry. Never the four-A student. Absolutely not."

Having initially harboured ambitions to be an actor himself, Sinclair went on to study drama at the Bristol Old Vic. Colman followed and took a job cleaning B&Bs to support him before deciding to enrol. "I do think it helped that I was so shit at everything else. If I had been good at science or a real wordsmith maybe it would have been harder to back myself."

She says her family life is "number one. Always has been. I just couldn't see the point without Ed or the kids." They have two sons, Hal, seven, and Finn, five, and their set-up seems to work; while Colman is on set, Sinclair retreats to the attic to write and is currently finishing the third draft of his first novel.

Right now Colman is bringing in most of the money ("We support each other. He'd do the same for me") but takes on mostly British projects so she can be at home. "I also genuinely care about British film and TV. I'll watch British stuff before anything else – it's a pact I made with myself."

She already has a number of projects lined up for the rest of the year: the C4 drama Run, The 7.39, and penned by One Day author David Nicholls for the BBC. In regards to whether she'll return to Broadchurch for the confirmed second series, she says: "I'm not allowed to tell you anything, so don't look at me. Don't look into my eyes."

Of the upcoming Baftas, she thinks it's "obvious" which of the two gongs she would most like to win. "Oh, the one for Accused," she says, looking a bit teary again. "Tyrannosaur and Accused just mean so much." She cannot articulate quite why, though. Is drama more difficult? "No. You can pour just as much energy into a comedy as a really harrowing role." Does she connect more emotionally with the dramatic work? "Oh, I don't know."

In the end, she says: "I just think the other girls up for best comedy performance deserve it more." I suggest dramatic roles are more revered by the critics. "Yeah, I s'pose. But I spent a long time telling myself awards don't matter. Of course they don't." And then you get nominated. "Yes, then you get nominated! And you wake up in the night having a sweat."