Olivia Poulet: Saying goodbye to Emma Messenger but staying in the thick of it

"It's a bit of a problem. My bosom is mentioned in nearly every scene," Olivia Poulet says, with a wry glance at her lack of cleavage. "In the script my bosom is constantly described as 'heaving', and characters are always shoving their face into my tits. Hopefully we'll manage with corsets and a bit of uplift here and there."

Best known as blonde Tory fixer Emma Messenger in Armando Iannucci's The Thick of It, Poulet, 34, is about to appear on stage at the National Theatre, in Ron Hutchinson's new English version of The Captain of Köpenick, opposite Antony Sher.

She plays Sissi, the voluptuous wife of the ineffectual Mayor. "I married a Teletubby – this hopeless dull rotund man – for money and power," Poulet says. "It's a sexless, loveless marriage."

Poulet has been building up a name for interesting theatre. In 2010, she was in Max Stafford-Clark's revival of Top Girls which came into the West End. Last year, she appeared in Philip Ridley's Shivered at Southwark Playhouse, as a single parent who experiments with sexual violence after her son is killed, followed by Penelope Skinner's thriller, Fred's Diner, at Chichester – "It was such brilliant dark, edgy new writing" – in which she played a 30-something on a belated gap year, in a wig of pink plaited dreads.

The Captain of Köpenick – at the 1,100-seat Olivier – will be the test. The original German play, written by Carl Zuckmayer, was first staged in Germany in 1931 as the Nazis gained power. Although it's set in 1910, the spectre of Hitler hovers over the production. It's also very funny and surreal. "One minute you're welling up and the next it's cartoon physical comedy," Poulet says. "It's really visual and epic. The set design will convey that crazy, hedonistic side of Berlin."

Unable to get a job or his papers, Sher's character, petty criminal, Voigt, picks up a military uniform in a fancy-dress shop, holds the Mayor to ransom, and confiscates the town's treasury.

Poulet finds it chilling that the Prussian cult of the uniform means the town is all-too-willing to obey an impostor's orders. "This idea of just following a man because he's in a uniform prefigures so much… I think audiences will have a very visceral response. It makes you uncomfortable, in a good way."

It is, she admits, "quite a boys' piece". "We've got a plethora of older men in the cast which is such a joy, it takes the heat out of the rehearsal room in a way."

But then, she did learn to hold her own with the boys on The Thick of It, including Roger Allam's Coalition minister and his weaselish bag-carrier, played by comedian Will Smith. "It was a brutal survival of the fittest with all of us biffing each other out of the way," she laughs.

One senses the penultimate episode of The Thick of It – with its mock Leveson inquiry – was good training for broad physical theatre.

"Absolutely. Joanna Scanlan [who plays Terri Coverley] got it so right when she said: 'It's like a pantomime curtain call for all the characters with Malcolm Tucker as the boo-hiss villain'. I could watch her all day. Her character Terri just tickles me with her Kindle and her Joanna Trollope and the way she never wants to be at work. I love the way she's oblivious to her unpopularity."

In person, Poulet is smart, funny, and not remotely jolly hockey sticks. Though she does have an eye for the absurd – immediately spotting a mousetrap under the table of the press office at the National Theatre. "So you're … ridden?" she says with a glint.

She grew up in south-west London and attended Putney High School, before studying drama at Manchester university. After performing with the National Youth Theatre she began acting professionally.

She had roles in Teachers and Outnumbered and then created quite a storm in 2005 playing the young Camilla Parker Bowles, opposite Laurence Fox as Prince Charles, in ITV's royal romance Whatever Love Means. She was very good as the sexy, horsey Camilla. Later, she played Carol Thatcher in Margaret, the C4 biopic about Margaret Thatcher starring Lindsay Duncan.

It doesn't take much for Poulet's patrician blonde good looks to turn cartoonish. She writes sketches and plays with comedy partner Sarah Solemani:they took Bird Flu Diaries and The Queef of Terence to the Edinburgh Fringe. They are now writing a comedy drama about women in their early thirties – "it's the next stage up from Girls" – for the BBC.

Like Miranda Hart and Catherine Tate, she knows it's important to generate your own characters. "It's a great relief when you start writing because it frees up so many things. You can write things you'd want to be in, and that you believe are important for others to see."

She has recently finished a screenplay, Row Girls, based on the true story of four girls who rowed across the Atlantic. "I think it could be quite beautiful actually, as well as funny and charming. There aren't many female sports movies. It's about ambition – what could possess four people to row 3,000 miles across the Atlantic?"

Back in 2010, Poulet had a cameo in Sherlock, in an episode called "The Blind Banker". At the time she was dating Benedict Cumberbatch. They met at Manchester university and ended their 12-year relationship in 2011. Both maintained a dignified silence about the split, though a year later he revealed his biggest disappointment was "not being a dad by the age of 32."

Today, Poulet is single and she is open about the pressure on thirty-something women. "We're the generation who have so much choice and freedom – and yet, ironically that choice and freedom can be suffocating," she acknowledges. "Technically you can have it all; you can work and have children, but somehow you can't make a choice. There are lots of things I want to do. You think: I can't pick so I can't jump. And actually I think sometimes, it's a case of leap and a net will appear."

She says the trigger for writing the BBC drama was the paradox of choice facing women in today's consumer society. "I was in the supermarket about to buy some cheese and there were six different cheddars and then I saw the Jarlsberg and Brie and I thought: 'Oh Christ, maybe I want that?' Finally, I left cheeseless because I couldn't choose. And that made me go: 'Something's wrong here.' "

Her new feminist bible is Caitlin Moran's How to Be a Woman. "I've decided it should be a book you finish and pass on to someone who is in need, so I've started doing that. There's one brilliant bit which resonated massively for me, where she says one of the deep, troublesome things about being a woman is that your creative fertility starts peaking in your mid-thirties, where people start taking you seriously as a writer or an actress or whatever, when just at the same time your fertility goes the other way. So," she sighs, "you're in this dichotomy of going; 'I just want three more years and then I can cement it down.' Whereas men can keep popping out bambinos until they're 70-80. It's terribly badly designed."

'The Captain of Köpenick', National Theatre, London SE1 (020 7452 3000) to 4 April