Daisy Lowe has got her knockers. Those people, for example, who suggest she’s only successful because of her parents. “Fair enough,” she says. “To people who say that, I say, ‘Sure. Go for it.’ Because if my parents weren’t famous I probably wouldn’t be doing this, to be honest.” She considers this. “But it’s also how I’ve managed to have a platform to be able to model with boobs and arse and some thighs, you know?”

Any discussion of Daisy’s appeal ends up at her breasts sooner or later. Vivienne Westwood calls her “a bombshell; a style icon for her generation, she looks super-sexy all the time”. Vogue editor Alexandra Shulman singled her out as a crusader against size zero, reversing fashion’s unhappy fixation with “jutting bones and no breasts or hips”, the current contentious topic as evidenced by last month’s hoo-ha over size 14 models at London Fashion Week. i-D editor Ben Reardon has had Daisy on his cover twice, the first time shot in profile, topless, by Terry Richardson with the barcode covering her nipple. “She’s not the thinnest, tallest or most beautiful model,” he says. “But she is stunning, sexy and inspiring in her confidence; a curvy sex-bomb. She’s the hot, cool girl at school who cut class and smoked ciggies behind upper school and pashed with the older boys. Plus,” he adds,

“I love the fact she’ll get naked at the drop of a hat.” And if striking a blow for feminism by flaunting your boobs sounds suspiciously like something Katie Price would come out with… well, it’s all a matter of context. “Daisy gets her tits out, but she does so successfully,” reckons her friend, TV presenter Alexa Chung, “because she’s cool.”

“All I’ve ever done is try to make girls feel better about themselves, because women are curvy,” Daisy says. “They’re designed for a child to come out here.” Well, quite. It’s the sort of thing she says. The same lack of self-editing that used to drive her ex-boyfriend, the music producer Mark Ronson, to distraction (“The amount of times he was like, ‘Daisy, stop opening your mouth! You cannot tell a journalist to piss off’”). That resulted in her revealing a cervical cancer scare to MTV (“They wouldn’t edit it out and I was, like, ‘Come on, I only told you because you asked me why I was looking so stressed’”). And, if her answers to my nosy questions about her private life are anything to go by, presumably she gives her publicist kittens on a weekly basis (ten minutes into our meeting, the gory details of a fiddly kidney/bladder operation are offered as the reason she’s returning to London after two years in New York. I’d have accepted “homesick”).

She wears her heart on her sleeve – Balenciaga today, a cropped jacket teamed with flowery Topshop skirt, tartan DMs and what can only be described as half a string vest (“Look! There’s bows on the back!”) – a 20-year-old, figure-of-eight-shaped chatterbox representing “real” women at precisely the time the waif has stumbled under her own Bambi legs.

Like all the best models, she’s mercurial: able to do vampy for Agent Provocateur, gal-about-town for Converse, demure for Pringle and cute for Marc by Marc Jacobs – becoming the face of fashion’s most unimpeachable label earlier this year. The whole package has won her a legion of adoring teenage fans – just this morning she was mobbed by “20 14-year-old girls” while buying a croissant on Carnaby Street. She’s part of the “überbrat pack” alongside fellow famous offspring Peaches and Pixie Geldof, Lizzy and Georgia Jagger; Jaime Winstone, Lily Allen and Coco Sumner. Not that she’ll thank you for the It-girl tag. “It annoys me sooo much. I work harder than any 20-year-old I’ve ever known,” she says. “I work my arse off.” And you believe her.

Daisy Lowe boasts not one, but three famous parents. Her stepdad is Supergrass drummer Danny Goffey; her biological father is former Bush singer Gavin Rossdale, something she only discovered six years ago via a DNA test. That came as a surprise to no one more than her mum, Britpop singer turned fabric and fashion designer Pearl Lowe. Then there was Rossdale’s wife Gwen Stefani – who reportedly was not best pleased. But Pearl was used to firefighting, having confessed to an evening of wife-swapping, and coming clean about her years of heroin and cocaine addiction, later written about in unstinting detail in her 2007 memoir All that Glitters, soon to be a film.

Today, Pearl and Daisy are chitchatting over a late mezze lunch in an upscale London hotel. Two nights ago, Daisy presented Giles Deacon with his GQ Designer of the Year Award at the Royal Opera House. Tonight, she’s walking for Chanel at Fashion’s Night Out, “a global celebration of fashion” organised by Vogue to tee-up London Fashion Week. She’s 11 days into giving up the American Spirit cigarettes, a healthy concession that’s naturally resulted in an appalling cold. (She waves her Vicks inhaler miserably.) Pearl, meanwhile, is admiring samples of the Christmas dress collection she’s designed for Peacocks, which Daisy models here – daughter being something of a mother’s muse. “I’ve worked with other models, but no one comes close,” she gushes. “She’s everything I wanted to be when I was younger. Taller, more beautiful…”

“Stop it!” bats Daisy.

“…skinnier…”

“Oh, shut up!”

“...thicker hair. I mean, everything. She has everything better than I do.”

Times recommends

• 12 party beauty tips
• Nutrition: is being big a genetic destiny?
• What it feels like: to pry into other people's lives