She’s meeting us today for afternoon tea in a London restaurant to talk about her role in the ITV drama Losing Gemma but she is also currently appearing on stage in Tom Stoppard’s play, Rock’n’Roll.

“I love being backstage with all the other girls in the cast,” she exclaims. “The cacophony of everyone talking together . . . ‘How are you?! Do you want some tea?! Oh God, someone’s crying!’ It’s like being back at school.” Miss Jean Brodie, of course, would have thought her the crème de la crème, though she emanates such a subtle air of self-satisfaction that you think more of the cat that got the cream.

But, hey, the daughter of the suave actor Trevor Eve and stunning Sharon Maughan (one half of the Nineties Gold Blend duo) is riding high on her own success. Apart from the play and the TV drama there are also two new films: Starter for Ten and Big Nothing. The latter also stars Simon Pegg and David Schwimmer. “The Schwimm was sooo cool,” she says.

India, on the other hand, where she filmed Losing Gemma this summer, was stiflingly hot. “We spent a lot of time filming in the bustling streets of Gurgaon, famous for its call centres. It was full of traffic, fumes and noise,” she says. “I’d been to India once before with a boyfriend. His Indian parents had a farm in the Punjab. We would wake up and there would be buffalo milk on the table, still warm from the udder! This was a much more urban experience.”

In the feature-length drama, Eve plays Esther, a beautiful but unassuming Northern girl who goes backpacking in India. Her companion Gemma (played by Rachel Leskovac) is a peroxide nightmare who lands them in all manner of trouble and weirdness after making friends with a dodgy couple played by Jason Flemyng and Koel Purie.

“Poor Esther is loyal and trustworthy. But she’s also vulnerable because her mum died recently and she has no family to turn to. It means she’s far more willing to tolerate the kind of behaviour from Gemma that I would walk away from.”

Eve, who normally plays the posh, blonde minx (as in The Rotters’ Club) wears her hair brown and adopts a Liverpudlian accent for the role. “The accent came quite naturally, perhaps because my mum is from that part of the world. When we watched Losing Gemma together the other night she had a tear in her eye. She said, ‘Darling, you’re just like I was at that age!’ And I do like to think that Esther is a little homage to my mum.”

She’s clearly close to both parents and recently returned to live with them in Chelsea after splitting from the actor Rafe Spall. Though rumoured to be dating her Rock’n’Roll co-star, Rufus Sewell, she comes from a family used to diverting nosy questions. “I’ll plead the Fifth Amendment on that one,” she smiles.

Her parents’ own marriage has at times come under public scrutiny. “A successful relationship takes a lot of work but I feel privileged that they are still very much together. After playing Esther who is, basically, a rootless orphan, I realised how lucky I was to have such a strong, tight family.”

As children, she and her younger brother shared their parents’ nomadic lifestyle. Before the arrival of her youngest brother, the family lived in Los Angeles for eight years. Eve recalls a swimming pool at their home shaped like California and hanging out with the rich valley girls. Paris Hilton went to the same school. “And what’s happening to those girls of my age is quite a phenomenon,” she clucks, disapprovingly. “They have just become commodities.”

When the family returned to Britain, Eve went to Bedales, the progressive public school, and later read English at Oxford. “My dad was always brutally realistic about the business and the fact that many people never find work. He felt that you should be fully prepared to do something else, and education was always the key.”

There was never much doubt, however, that Eve would follow in her parents’ footsteps. She remembers being “utterly devastated” at primary school when other girls in her class were picked to be beautiful fairies in the Christmas play and she was made to play . . . the tree. “I had to stand at the back singing ‘Oh, Christmas Tree’. It felt like the worst thing that had ever happened to me, but I suppose it made me all the more determined.”

At 15, she narrowly missed out to Scarlett Johansson when Robert Redford was casting The Horse Whisperer. Since then, she has been making her way gradually into the public arena, rather than being fired into it, like Johansson. Even now, she is not generally recognised in public, but is philosophical about her impending fame.

“I don’t think it’s that hard,” she says confidently. “I mean, I’m not Kylie and I wouldn’t want to discredit the problems she has in getting out of the house in the morning, but I think fame is a perfectly fair price to pay for the privilege of doing the thing that you love.”

We’ll remind her she said that.

Losing Gemma, Mon & Tues, ITV1, 9pm

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