Lily Cole: The catwalk queen who conquered Hollywood

If you don't count the legions of magazine covers that she has graced, I have encountered Lily Cole twice before. The first time was in 2006, when she was sitting on a podium next to the iconoclastic Marilyn Manson to announce his (as yet still unrealised) directorial debut, Phantasmagoria: The Visions of Lewis Carroll. Just about anyone seated next to the Goth rocker in full make-up would probably pale into insignificance, but a demure-looking Cole – just 17 at the time – held her own. Not that this was a surprise. Since she broke into the fashion world as one of the industry's new faces in 2003, she's modelled for just about every major designer worth getting out of bed for.

My second encounter was no less glamorous. Proving she wasn't above mixing with the masses, she was standing near me at a Radiohead concert in London's Victoria Park last year, accompanied by the slightly less iconoclastic Jude Law (who got a sharp telling off from one fan for talking through the Kid A track "The National Anthem"). "How funny," murmurs Cole, when I explain to her the second of these sightings. Admittedly, there's a chance she must think I'm stalking her, not least when I enquire what her dress is at the end of our interview (it's a slick navy pinafore number by Chanel, by the way) and she mistakenly believes I've asked for her address.

Still, she must be used to being approached by strange men by now. When she was 14, Cole was munching on a hamburger and fries in Covent Garden when a scout-turned-actor named Benjamin Hart approached with the intention of convincing her to undertake a photo-shoot. Believing him "dodgy", she refused, as most sensible, well-heeled girls should do, though later relented. Before long, she was on the books at Storm, the élite modelling agency that launched the likes of Kate Moss, Cindy Crawford and Sophie Dahl. And when Steven Meisel, the photographer behind Madonna's Sex book, snapped her for Italian Vogue (the first of seven international covers for the magazine brand), Cole's future was assured.

So far, so fairy tale. Alongside the likes of Australian Gemma Ward and Canadian Heather Marks, Cole's distinct look – the cascade of red hair, piercing blue eyes and facial features that recall "a porcelain doll having a fit of the sulks" as one writer put it – suddenly became oh-so-desirable for fashionistas everywhere. Campaigns for Chanel, Christian Lacroix, Longchamp, Cacharel and Hermès and catwalk appearances for DKNY, Versace, Jean Paul Gaultier, Alexander McQueen and Louis Vuitton took her fortune to an estimated £6 million. In 2004, she was named Model of the Year at the British Fashion Awards, the same year she appeared with Ward on the cover of British Vogue above the headline "The New Pretty".

Of course, "pretty" doesn't begin to describe Lily Cole. Terry Gilliam does it better. The film director cast Cole in The Imaginarium of Doctor Parnassus, one of two films she has out this autumn. "I just wanted somebody who was extraordinary looking to begin with," he says. There's no question he found the right girl. With a willowy beauty more in common with the pre-Raphaelites than the waif-like clothes-horses of today, her slender (but not worryingly so) frame is supported by a pair of legs that lend her the appearance of an exotic bird. As such, when all 5ft 10in of her enters the hotel suite where we meet, she does not so much walk into the room as step in like a graceful stork.

What's fascinating about Cole, now 20, is how beauty is just part of the package. Back in 2006, the press went bananas when she announced she was planning to put her career on hold to enter Cambridge University. "I think that a lot of stuff that gets into the news nowadays is not very important," she reflects now. "It is quite shocking the amount of interest I got from choosing to go to university. And that people would be surprised that I'd have any incentive to want to learn more. Essentially that's why people are surprised. You have a career, you have money ... why would you? Of course, it doesn't take a very interesting person to realise there are more reasons why you want to study."

Admittedly, it's a delicious image, a supermodel slumming it in halls, shooting pool and supping snakebite in the student-union bar. Not that there's much chance of this. Cole scored three As in her A-levels (in English, politics and philosophy-and-ethics) and is evidently something of a bookworm. While she initially gained a place to study social and political sciences at King's College, she is now reading history of art there. I ask what made her choose that. "Because I don't know a lot about it," she shoots back. "I like doing things I haven't learnt about yet. I've always been interested in art, and I love doing art ... like painting and stuff. So it's nice to know the background behind it. I'm interested in humanities and histories, and it covers a lot of that as well."

Cole has her mother, Patience Owen, to thank for this. Born in Devon, Cole was raised with her older sister Eveline in London, where Patience encouraged her children to embrace the arts. While little is known about the girls' father (rumour has it he was a fisherman from Brixham), Cole is evidently tight with her clan. "Me, my mother and my sister are very close," she told one reporter recently, "and I don't think they would ever let me be anything other than grounded." Her upbringing has left her with a sound morality, too. Adopting the likes of Water Aid and the Environmental Justice Foundation as causes during her modelling days, she signed up to M&S because of its Plan A green initiative and even cut her relationship with De Beers, because of the diamond company's treatment of native bushmen.

But if this makes her sound worthier than thou – which she's not – she does boast a good sense of humour about herself. Her first acting role was in St Trinian's, the 2007 update of the black-and-white comedies about an anarchic all-girls school, where she played Polly, one of the more studious characters, who hides behind her books and John Lennon spectacles. "It was fun," she smiles. "There wasn't really very much pressure. I didn't really take it very seriously. I wasn't really committed to the character. But it was a good way to get used to the technicalities of being on a film set, and not feeling like an impostor. And also I liked the idea of being a model and playing a geek. I quite liked taking the piss out of the fact I do study a lot. It was just a bit sardonic."

Cambridge is evidently a good place for Cole right now, being surrounded by all those high achievers. She hasn't been treated any differently, she says. "They're all really smart, so I think they've got a good perspective as well on my normality, so it's been fine." For the most part, she has put her modelling career on hold to concentrate on college, something she didn't do during her A-levels when she used to complete homework assignments while on planes ferrying her to and from the catwalks of Milan, Paris and New York. "That was really hard work," she sighs. So how will she approach it now? "Just play it by ear," she shrugs. "If work becomes too much, I'll make that decision at the right time, if I can't balance both."

For the moment, it's acting that interests her more than modelling. After St Trinian's, she was approached by Sally Potter, the sometime-visionary director behind Orlando, to appear in Rage. The conceit sees a student named Michelangelo shoot a series of interviews on his mobile phone for a school project, while backstage at a fashion show by a top designer named Merlin. The interviewer is neither seen nor heard, which leaves his subjects to pontificate to camera. These include: Dame Judi Dench as an acerbic dope-smoking fashion critic; Jude Law as a transsexual model named Minx; Steve Buscemi as a cynical fashion photographer; Eddie Izzard as a shady owner of a fashion house; Dianne Wiest as his old-school brand manager; and John Leguizamo as his stupidly loyal bodyguard.

With each character shot against just a plain, brightly coloured backdrop, the film suffers in that it rather feels like an experimental theatre piece gone awry. Indeed, when it premiered at the Berlin Film Festival earlier in the year, it was met with scathing reviews by the trade papers, who dubbed it amateurish and self-indulgent. But while it might be too marginal a work for cinemas, it's now being released over the internet through Babelgum, an innovative web-based platform for streaming brand new film and music content. What's more, in a landmark move, it will first be released episodically, in seven segments over the course of a week, via Babelgum's free mobile application for iPhones and the like.

Certainly, there's something appealing about sitting on the bus and watching Lily Cole pop up on your phone. She plays Lettuce Leaf, a model ("New Face of the Year – last year", as she tells us) who, like the others, must deal with the fact that two of her peers are killed off-screen during the week that these interviews take place. If Potter's attempt to fuse thriller elements with a fashion- industry satire is rather wayward, early reviews all agreed that she got one thing right. "The big revelation here, however, is Cole, who shows thesping muscles one wouldn't have expected she had on the evidence of her previous turn in St Trinian's," said Variety. While Screen International praised Cole's "dramatic bravura".

Lettuce is a fragile soul and Cole invests her character with a poignancy that's missing from many of the other performances. "I think one of the key themes with her is being very young and grappling with issues of loneliness, in this adult industry that she's not necessarily chosen to be in, but has found herself in," she says. "And I'm sure that is familiar to models, because a lot of them do start very young, away from home." So does she recognise herself in Lettuce? "Not in her character so much but the emotions that it creates, yeah." Cole then makes a salient point. "Unlike most jobs in the world, models don't aspire, usually, to become models. You just get plucked from all corners. And so there's not a shared sentiment or interest that's taken them there. They just happen to fit the aesthetic."

Oddly, Cole first auditioned for the role of Minx, which ended up being played by her friend Jude Law (while press reports have romantically linked her with Law, as well as Bryan Ferry and Colin Farrell, she's currently stepping out with Enrique Murciano, the 36-year-old American star of CBS series Without a Trace). So did Law help her get the role? She shakes her head. "It was very curious when I found out he was playing the other role that I'd auditioned for. Obviously, they'd imagined originally putting a woman in that role." She discovered Law's involvement while they were both on the set of Gilliam's film. "I'd already been committed to Sally's film for a while longer, and then he just had been asked and decided to do it. And then he found out I was involved."

Law's involvement with The Imaginarium of Doctor Parnassus, a dark morality tale set in contemporary London, is tinged with sadness. Along with Farrell and Johnny Depp, Law stepped in to help Gilliam complete the film when the Australian actor Heath Ledger died in January 2008, midway through production. With each replacing Ledger at various points in the film's CGI fantasy "imaginarium" sequences, the plot concerns an ageing conjuror (Christopher Plummer) who tours the capital luring unsuspecting souls on to his wagon and into his actual imagination. Parnassus has made a pact with the devil for immortality, and Satan (Tom Waits) comes to collect payment in the shape of Parnassus's daughter Valentina (Cole), who becomes his property when she turns 16.

At this point, Cole's usual eloquent flow of words dries up. Unsurprisingly, the death of Ledger hit her – and everyone else – hard. "I don't want to talk at length about it," she says, "but it was just a very difficult thing for everybody involved. I really admire Terry, and the director of photography, and the producers, and the actors ... everybody involved who managed to complete the film and do it for the right reasons. They all knew Heath and they all loved Heath. They wanted to honour it [the production] for the film's sake and finish it also for Heath's sake. And they worked very hard to pull together in difficult circumstances, to see that it did get finished."

While most will flock to the film for the curiosity value of watching Ledger's last role, it again shows Cole as an actress of great potential. She admits she was "terrified" of doing the film but thinks it's good to face your fears. "I hope I don't lose that," she says. Understandably, she detests the phrase "model-turned-actress", not least because it doesn't apply to her. She first took up acting during school, before she ever strode down catwalks for a living. "I was super-keen on it when I was younger. But I never really thought about pursuing it professionally, because it seems like such an intangible thing to try and grasp after. I don't like chasing things." Indeed, much as she fell into modelling, she fell into acting too – with roles simply being offered to her out of the blue.

Currently shooting a small role in Roland Joffé's Spanish Civil War-set tale There Be Dragons, university studies permitting, she's also still attached to Manson's Lewis Carroll film, in which she will play Alice. She admits she has no idea if it will happen. And acting, just like modelling, may not even interest her by then. "I could feasibly see myself completely changing direction," she says, citing Woody Allen's Annie Hall to illustrate her point. "There's a good line in there about relationships. It goes, 'A relationship is like a shark. You've got to keep moving or it will sink.' Life is like that. With a career, you have to keep changing and being interested in what you're doing." So far, she's done this perfectly.

'Rage' is available online at babelgum.com, and on DVD from 28 September. 'The Imaginarium of Doctor Parnassus' opens in cinemas on 16 October