Johnny Depp and me: Vanessa Paradis on acting, gigging and life after a break up

There's the shared childcare – she and Johnny Depp, from whom she split in 2012 after 14 years together, divide custody of their children, Lily-Rose, 15 this month, and Jack, 12, and her life is a constant blur of transatlantic shuttling. There are the cigarette papers and pouch of tobacco she's currently balancing in her lap as she attempts to build a cigarette.

And then there are the three careers.

Good thing the 41-year-old Frenchwoman has a hefty quarter-century's experience of taking care of multifaceted business, ever since "Joe le Taxi" shot the then-14-year-old to international fame. Judging by her demeanour – relaxed, scruffy-chic, sweary, smart – Paradis isn't a woman fazed by much. She is, it seems, the epitome of Gallic sangfroid. Twenty-five years in the spotlight can do that for a woman.

First off, the actor Paradis' new film, Fading Gigolo, is out this weekend, and there's promotion to be done. It's set among the Hasidic community in Brooklyn, also stars Sofia Vergara and Sharon Stone, and is a lightly comic story about an unlikely male prostitute (John Turturro). Paradis, in her first English-speaking role, plays a widow employed as a "lice lady" – a woman who makes a living removing nits from kids' hair. Talking about the film, the enthusiastic words tumble out of her, all accented and emphasised and delighted.

"Woody Allen is just like you see in his movies," she says of her other co-star in the film, which is written and directed by Turturro. "He's an improviser. And he's such a funny improviser!" She tried hard not to be distracted by Allen's comic gifts, all the while maintaining "a semi-accent. Not New York – Turturro said he didn't really want to know where I came from, whether it was Israel or a little French. So I mixed them both, and I'm not the best at accents!" she admits with an embarrassed grin. "So I'll be surprised to see my accent performance."

Then there's her music: next month the singer's current album, Love Songs, is receiving another push with a one-off UK show at the Forum in London.

"It just makes sense," Paradis shrugs, perfectly Frenchly, of her enthusiasm for promoting her sixth studio album one year into its shelf-life. "You make a record then you go and play it for the people. Plus, you enjoy it so much! It's hard work, lots of travelling, being in shape whether you come out of a plane or a train or hundreds of miles of bus… But then comes the time for the show, everybody puts on their clothes, lights on, boom: you sing. It's quite amazing, you know?

"And it's risky. Some days you wake up, you don't have your voice, you're tired, you're this, you're that. But at night for two hours, ooh, you just push the adrenalin to the max.

"I'm so admiring of musicians," she continues, "and just the way their instrument is like another part of their body. Then the community life with the crew, the sound, the light, everybody going on the road together. The band are new musicians for me but the crew is the same I've been working with for a few tours, so it's just nice to be together – travelling, singing, making music," she concludes with a flash of that famous gap-toothed smile.

Finally, there's the modelling: Paradis has an ongoing, 23-year relationship with Karl Lagerfeld and Chanel. It must be a particularly meaningful partnership, I suggest, given the fickle nature of fashion. Paradis – the picture of Parisienne elegance, but also a relaxed, rolly-up-smoker – pauses between licky dabs at her cigarette paper.

"Yeah, yeah, there's that," she agrees, tentatively. "But we are very replaceable, you know? Cos fashion needs new people. Cinema, I guess, also," she adds in her occasionally back-to-front French-accented English; she sounds like a lyrical Yoda. "Music is different – music, I guess, is more protective. Obviously you dream of somebody's voice, but also it's a lot about what the music does to you. But fashion…" she muses, "it's less of a free world. It has to do with what people want to see, who they want right now."

Still, Paradis has been consistently in demand since adolescence. Family connections led this daughter of successful interior designers k into a recording studio, initially for a Junior Eurovision contest, then to sing the song that would take her to Top of the Pops in 1987, and then around the world. But she says that even after that first flush of pop success, she wasn't sure she wanted music to be her future.

"It wasn't that clear. I started to love music at the same time that I loved cinema, which was through musicals, the old MGM musicals with Gene Kelly. I was completely mesmerised and obsessed. I would watch those movies and sing along in English before I could speak English cos I had learnt it from those films. Music, cinema, dance, Technicolor – it was such a perfect world for me. It gave me goosebumps," she says with another toothy beam.

Her own daughter is now at the age she was when she started out, and it seems she might be following her mother's lead: Lily-Rose has a songwriting credit on Love Songs, alongside her father.

"This song, 'New Year', is actually really old – it was composed when Lily-Rose was six! I couldn't find a melody that I liked but at one point she started to sing this thing over the chords! She sang not only the first phrase with the melody but even the words.

"And I was just like, 'Oh my God, what's that?' It was like a melody that can lighten you up in church. It was so beautiful. And I never forgot that melody. Then I finished the song – but the song would never have been that if she hadn't found that first, perfect phrase."

Depp, meanwhile, had come up with those chords, making it a proper, three-way, familial collaboration. "Yes," she beams. "But it took a long time. Johnny's chords were really so pretty. And I knew I wanted to make a song of it, but it took all those years. I even tried it on my last album," she says of 2007's Divinidylle, "but I couldn't find it. I needed Lily-Rose's melody."

Her daughter, she says as an aside, is a "non-stop" singer. But despite her mother's teenage example, Lily-Rose is seemingly yet to request her own moment in the spotlight. As the daughter of internationally famous parents, the adolescent Ms Depp-Paradis must be as aware as anyone of the likely pressures. Mum, too, is ever-alert to the spotlight that will fall on anything her children do.

"I started so young. I don't regret that at all and I really thank my parents for letting me do it. But at the same time, back then it was different – not that I'm 150 years old!" Paradis laughs. "And not that it was all safe back in the late 1980s. And now, being a mom, I don't know that I would have said yes," she admits.

With regards to Depp, Paradis remains as tight-lipped as she was when they were together. The 50-year-old's recent engagement to Amber Heard, 28, his co-star in The Rum Diary (2011), will have only stiffened that resolve. But when I met her in Paris last year, I asked her about Love Songs, and the lyrics of the title track ("I don't know nothing about love, you know"). Observers might reasonably guess that the album was a product of her split from Depp.

"No," she smiled in reply. "And I didn't even write that lyric, so I am all safe. I would never do an album as a journal. A journal is yours, but music is something to share with people. Everything that has been said or written in the record are universal ideas that people can relate to and will hopefully make them feel good. And I am completely protected there," she added with another smile, "[because] I didn't write the songs."

Beyond that, no, Vanessa Paradis will not be saying any more about her ex, nor indeed much about her private life. Yes, her children are at school in America, "but I don't really wanna discuss that". And no, she's not planning to relocate to Los Angeles. "I know my Paris friends get upset with the weather and the traffic and the strikes and this and that. But because I don't have to live it every day, you know, it's raining and I applaud! I'm so happy! I don't care about the bad moods of the taxi drivers. It's just part of the city."

Anyway, she can make her music and her films anywhere in the world: she's just back from shooting another project with Turturro, a short for the portmanteau film Rio, I Love You, and the American can't speak highly enough of her. "Vanessa just disappeared into the Fading Gigolo role with the wig, the research, the dress," Turturro tells me. "She far exceeded what I thought we were going to capture. Her presence changed the whole movie – she gives it a depth and gracefulness and a delicacy.

"You can act certain things," he adds, "but some of who she is is just very specific and very rare. You just don't see people like her – she's strong but delicate. And for someone who's been famous since she was a kid, Vanessa is so unspoilt and aware of everybody else around her. I adored working with her. Woody didn't know her, and he thought she was Hasidic Jewish! I had to tell him, no, Woody, I don't think she's even Jewish!"

Clearly this just-turned-41 and not-long- single woman is a pro. But more than that: as Turturro attests, she's a natural. You can tell as much from her preferred look. For all the high-glamour of her photoshoots – she was on the cover of the April edition of French Elle – her natural style is very much boho rock'n'roll.

"Yes!" she nods with smiley emphasis. "And you know what: even when it's chic, I want to be comfortable. You're walking or sitting and you're in something itchy and tight all night? You're not gonna have a good evening!" she shouts. "I wear something uncomfortable for a picture, because I know it won't last. But what's the point in going to one of those dinners or ceremonies or whatever and being just like, eurgh," Vanessa Paradis grimaces, "tortured?"