Never mind the Sex Pistols... it's Hollie Cook

She’s the daughter of Johnny Rotten’s drummer, but Hollie Cook is more Lily Allen than Anarchy in the UK

Hollie Cook hoicks up the sleeve of her watermelon-patterned blouse to reveal, on her right forearm, a tattoo with the word “grandad” inscribed, in memory of her father’s father, who died last year. She offers a faraway smile. “We were very close.” Since his death, she has moved back into his house in Hammersmith. It’s the same house she grew up in with her parents, Sex Pistols drummer Paul Cook and former Culture Club backing singer Jeni Cook, who themselves now live just a few doors down, the whole clan loyal to this small corner of West London. Hollie says she feels closer to her grandfather this way, amongst his furniture, much of which he made himself (he was a carpenter). “Plus,” she says, “there’s the cat to look after.”

Hollie Cook is 28 years old, but could pass for 18. Her cherubic face, out of which peer two sheepish eyes, is framed within an explosion of frizzy, untameable hair. She looks every inch the modern pop star, but doesn’t necessarily sound like one. Her second album, Twice, the follow-up to 2012’s much-lauded eponymous debut, ploughs the kind of summery pop/reggae groove that requires a lot of sunshine to work most effectively. It is preposterously laid back, occasionally reminiscent of Lily Allen, and reminds listeners of Lindy Layton, the young woman who, in 1990, provided the vocals for “Dub Be Good to Me”, the No 1 hit from Norman Cook’s Beats International project.

“Always loved reggae,” she beams, revealing teeth Colgate would kill to endorse. “I took a little break from it in my mid-teenage years, when I got angry and moody and just wanted to listen to acts like The Deftones and Korn, but I soon went back to it.” She grew up in an inescapably musical environment – Edwyn Collins was a family friend, Boy George her godfather – and by the time she was 10, her dad was gainfully re-employed as a Sex Pistol for what would become a very lucrative series of reunion tours.

“I remember seeing them at Finsbury Park in 1996, and finally realising what all the fuss was about,” she says. “They were great, and backstage was very exciting. Though I do remember [support act] Iggy Pop wearing plastic, see-through trousers, and no pants. I was like, ‘o-kay...’”

Her interest in dad was far less pronounced than everybody else’s around her. And so she did not become obsessed with punk herself: throughout her surly teen phase, she’d sooner reach for something by the Buzzcocks than she would Never Mind the Bollocks. She’s never watched any punk documentaries, nor read any of the countless punk biographies.

“I never felt I needed to,” she reasons. “If I ever had any questions, I just asked him.” And did she have a lot of questions? She shrugs. “Some.” This carefully maintained lack of interest means that Cook never really fought – unlike the offspring of so many famous parents – to escape his shadow.

“Oh, I never burdened myself with anything like that, and it never fazed me. I mean, it’s not my fault I was born to the drummer of the Sex Pistols, is it? I’m just proud I have such a cool dad. Though it was always a little awkward when teachers asked me what he did…”

The cool factor, however, brought problems. “I’m aware that he has already done something way cooler than I will ever be able to do. That’s a little frustrating, I suppose. But, you know, oh well, never mind.” By her own admission, Cook was a precocious child. She went to a performing-arts school and, later, to music college. “I was always singing, dancing, demanding an audience.” Did she have her father’s rebellious streak?

“Not really, no. I was horribly shy and massively introverted for a while, hated school. Who knows why? Hormones! And then I suppose I became a typical teenager, rude and horrible to my parents, almost as a rite of passage. Though I never felt I needed to be a complete arsehole. I was never a total bitch.” But then there wasn’t, she admits, much to rebel against. “There wasn’t a whole lot of discipline going on at home, but they did bring me up well, and I have good manners.” It’s true, she does. “The only thing I wanted to do, really, was go to gigs, and they could hardly stop me doing that, now could they?”

While she was at music college one of her father’s friends, Ari Up, from all-girl punk act The Slits, reforming in 2007, asked Hollie to sing backing vocals. She left her course, and spent the next four years touring with the band, leaving only in 2011 to focus on her own music. Another family friend, Ian Brown, then invited her to open for the Stone Roses in 2012, in front of a crowd of 75,000. A baptism by fire, surely?

She laughs. “You know what? I remember none of it! Funny how the brain works, isn’t it? But my mum did film the whole thing, and I have watched it since. It looks like I was enjoying myself.”

Her mother is a quiet, but persistent, influence in her life. Jeni Cook quit music years ago in favour of retraining as a nutritionist, and raised her daughter a strict vegetarian, encouraging her to avoid any sort of impurity.

“I did smoke, which she wasn’t happy about, so she was very relieved when I gave up. But with diet, I’ve learned that you have to listen to what your own body tells you, not anybody else.” And what does her body tell her? “It tells me that, sometimes, I just want to eat steak, you know?”

‘Twice’ by Hollie Cook is released on 12 May