A good thing too, since Atlanta has now decided to move to London in order to build herself a career. She’s not entirely sure what she wants to do, but so far she’s modelled for Net-a-Porter, blogged about London Fashion Week for Elle magazine and DJ’d at a celebrity-fuelled Topshop party. Did she play any Duran Duran? “Girls on Film is in there. The songs that I love, I love. But I don’t love it all because it’s my dad, and I don’t hate it all because it’s my dad.”

Right on cue, Taylor phones for a chat. Still the band’s bassist, he is now married to Gela Nash, co-founder of Juicy Couture. Atlanta is off to stay with them at their Elizabethan mansion in Wiltshire, hence all that luggage. “My dad and I are going horse-riding on Sunday,” she says. “I’m a little scared because he isn’t very good. The last time we went riding, he freaked out his horse, and my horse, which went off across the field … He doesn’t look right on a horse.”

In other ways, Taylor clearly hits the spot as a parent. She pulls up the leg of her jeans to show me the sparkly silver socks she pinched from him. “Actually, he has a really good sock collection.”

Well, of course he does! He’s John Taylor! Having been brought up in LA, she doesn’t get what that means to a child of the Eighties like me.  “I still don’t fully understand, and I don’t know if I ever will,” she admits. “Maybe it would be different if I grew up here.” She says she only realised the extent of her father’s fame 10 years ago in Rome. “They had to have security guards outside the hotel and we had people following us. It was crazy. And when I went on tour with him, my Twitter account was blowing up because all these Duran fans were tweeting at me. I’d post a picture of me, and my dad’s foot would be in the corner, and they’d be saying: ‘Oh my God, is that John Taylor’s foot?’ She laughs heartily.

Perhaps she’ll understand better when she gets around to reading his just-published autobiography, In the Pleasure Groove. But will she be able to cope with those dodgy details about the sex and drugs? “He told me this was the mild version!” she protests. “Oh man, am I going to find out I’m actually someone else’s daughter?” No, but there is a sordid episode when her father discovers he’s got an STD, I say. “Basically, I wouldn’t be surprised by anything — I know he was a really good-looking dude in a super-cool band,” she says calmly

Atlanta was brought up by her mother after her parents’ split. De Cadenet is now married to Strokes guitarist Nick Valensi, with whom she has six-year-old twins Silvan and Ella, and has reinvented herself as a celebrity photographer and TV interviewer. Understandably, she was determined that Atlanta wouldn’t follow in her own teenage footsteps. “I wasn’t allowed to dye my hair or pierce my ears till I was 14 — most people were doing that at 12,” says Atlanta.

“My mum was really specific about certain things. I wasn’t allowed to eat fast food — I was always eating egg salad sandwiches. The other kids would have their mums drop off In-N-Out [burgers] and I’d be there with my egg salad on wheat toast going, ‘Please — a French fry?’ They’d be like, ‘Egg salad girl, get out of here!’ When I was in high school, I was not a cool girl,” she concludes.

That’s not to say Atlanta didn’t rebel in her turn. She has refused to go to college, hence her arrival in London. And at school, she didn’t conform to the norm. “I used to wear only floor-length vintage peasant dresses,” she explains. “I went to a nice private school. It was unusual to be wearing vintage at that point.

“I got really into wearing my mom’s clothes. She had this old pair of high-waisted shorts that I loved. That wasn’t cool at the time. I was a loner — I didn’t really have friends at school, and most of my friends went to the art schools.”

Atlanta’s long-term boyfriend is artist Mike Gittes; they have been dating for two years and live together when she’s in LA. He’ll be flying over to see her soon but right now she just wants to concentrate on her career. “That’s what I like about being here — I don’t have any distractions,” she says. “I’m happy to go to bed early, wake up early, look nice … Now I’m 20, I feel like a grandma.” Oh dear. What will Mother say?