What would be the right setting in which to interview the queen of English rock? A gothic ruin? In the crook of a tree, perhaps, in an ancient oak forest? On the set of Miss Havisham's cobwebby dining room? All would seem more appropriate, certainly, than the banal, style-free zone of a room in the BBC's London HQ, in which Florence Welch looks as implausible as an antelope in an office supplies showroom.

She's six foot tall in her platforms (five foot nine barefoot, she tells me), with her familiar pale face and blast of red hair, wearing a characteristically amazing get-up of a figure-hugging, ankle-length polo-necked dress in a print of yellow and brown, scudding autumnal leaves, with a matching shirt. She resembles one of Narnia's talking trees or, more accurately, a whispering tree since her voice is, to start with, so very low and her manner demure, almost timid.

It takes her some time, for instance, to meet my eye (she has a steady, olive green gaze) - preferring to look at her appealing manager, Hannah Giannoulis, who is sitting behind me. The singer has rings on all her long, thin, alabaster fingers, one of which bears a tattoo. Her bracelet resembles a medieval torture-chamber wrist whacker. After about ten minutes, she stops jiggling, relaxes and you sense her whole body slowing down as her thoughts speed up.

Florence speaks in the rhythm of a song - a breathy, staccato gallop, almost a rap, where the words are one long, drawn-out babble, framed by ethereal, melodic waves of conversation. She also talks as an abstract painter paints, conveying information in a way that is totally precise (in that you know exactly what she means) but highly unusual, even poetic. She's one of the most genuinely original people I've interviewed.

When she tells you about life on the road, you get a real sense of the strangeness of being a global rock star; the adrenalin of performing followed by the lonely comedown in hotel room after hotel room; as though your whole life were one endless business trip. "Night times are really hard," she says. "You can't even really read on tour because I can't concentrate and I'm obsessed by books. I always worry that touring is making me stupider… so I try to stay engaged."

"It is strange to be out there and you're never really alone. You're in front of so many people and you're connected on such a grand scale. But you miss actual intimacy and home and connection. You start living for the gigs and the stage becomes the only normal place because everything else changes but the stage seems to stay the same. And then the stage becomes your whole existence, and that's slightly terrifying as well."

I love her description of the effect on her London home of the relentlessness of being away: "My house is like a graveyard of suitcases. I'm so sick of it. They haunt me they're these big black boxes which I have to fill, refill, empty. It's just constant. I'm actually a homebody. I very much like to potter and just be. I'll probably end up building a house for myself out of suitcases."

One of the ways she tries to stay engaged while touring is to go to art galleries - "What is that amazing Louise Bourgeois painting that says, 'Art is a guarantee of sanity' - I like that a lot. She also did a series of napkins that said, 'I've been to Hell and back and let me tell you it was wonderful'; it was stitched on these really beautiful, quite twee blue napkins. I love Louise Bourgeois.

She has a strong sense of aesthetics, as one might guess from her singular dress sense. When she was an art student, at Camberwell (she left after 18 months to pursue music), she describes herself as an installationist: "I'm obsessed with creating environments and the atmosphere of places and whether the place feels good. I'm definitely a sensory control freak. I'm completely chaotic with dates, times, maths but when it comes to what things look like, taste like, sound like, feel… I'm very, very particular."

Her mother, Evelyn, a Renaissance art historian, comes from a long line of East Coast American academics; her father, Nick, was an advertising executive and is now leading a rather different life as the "maitre d", as his daughter puts it, quaintly, "of an organic campsite in Tonbridge". (He was also her tour manager at the start of her career.) Her parents split up when Florence was in her early teens, and she is still - at 25 - living at home with her mother ("she's desperate for me to leave"), although she has recently bought a house in the same neighbourhood in south London.

Her paternal grandfather was Colin Welch, former deputy editor of The Daily Telegraph; her aunt, Frances, is married to humorist and journalist Craig Brown. It's debatable whether coming from a family of journalists and allied media would make you more guarded about exposing yourself to the press or less so. Certainly, in Florence's case it appears to be the latter.

Early interviews, especially with music magazines, after the phenomenal success of her first album, Lungs (one of the bestsellers of 2009 and 2010), were terribly naked; she talked of her anxiety and depression, excessive reliance on alcohol and the drag of hangovers. But she restricted her tales of heartbreak with her on-again off-again boyfriend to autobiographical songs.

I ask her if she has a boyfriend now and she answers, "Yes, I do… he's here somewhere," and starts looking under her chair and behind the sofa. Now, Florence, is your boyfriend perhaps an imaginary boyfriend, a Tom Thumb-size paramour? (She's been reading Angela Carter's recasting of fairy tales, since seeing a review that mentioned the late brilliant writer and the young singer/songwriter in the same breath.) "He's always disappearing," she says. "He's such a wanderer. Such a daydreamer." (I think this means that he's real.)

We talk about the process of songwriting. I mention something Nick Cave said to me about his love songs waiting patiently for him to finish them. "His was one of the first gigs I ever went to," Florence says, which will probably make him feel terribly old. "My parents took me. Marianne Faithfull said something like, 'Be careful what you write because it's almost as though you start willing it to happen.' She was talking about when she was about 16 and wrote Sister Morphine [later recorded by the Rolling Stones] and then she went on to have problems with drugs."

"It's funny because a lot of my songs almost seem to be about the breakdown of things. I think I'm slightly a chaos merchant like, always waiting for the other shoe to drop off."

Is she a positive person? Florence turns to her manager, "Am I?" "I think you are, apart from with yourself," Hannah says. "I can be quite Doomsday to myself, Florence admits. But I don't think I'm a negative person. I find a lot of joy in things. I get very, very excited."

She admits that some of her songs are pretty dark and tells a funny story about her dad: "I did a cover of a song by the band Beirut and my dad loved it. He thought I'd written it and he said, 'It's really nice to hear you write something upbeat.' I had to say, 'No, that wasn't me'." 

"My mum does worry about me a little bit listening to my songs, like Shake It Out. (Sample lyrics: "Regrets collect like old friends/Here to relive your darkest moments/I can see no way, I can see no way/And all of the ghouls come out to play…" "And I'm damned if I do and I'm damned if I don't/So here's to drinks in the dark, at the end of my rope…")

The funny thing is that you can see Florence singing this, solo, with a spare acoustic and (never mind her mother) you find yourself worrying for her; the song and her singing of it reek of despair at dawn. But then you see a clip of her belting it out in a mini-dress of sequins that makes her look like the Empire State Building, with a grin on her face, and a backing choir of male singers, and it sounds completely jubilant.

Florence says her songs now are a lot to do with battles with myself. I think when you're 25, you're coming to terms with not being a teenager any more and what kind of adult you're becoming - your ideas about morality and relationships. It's always, like, my chaos and control thing. I'm actually a control freak but try to keep things together. So I'm trying to find a balance in my life but right now it seems to be all or nothing."

I'm surprised to hear, at first, that she has a stylist, Aldene Johnson, who found her autumn leaves ensemble. It's mainly for the stage and stuff. We met four years ago when I got the Brits thing [Lungs won the British Album award in 2010] and we've been together since. Florence didn't have a clue about how to dress for the red carpet and now she's having to become used to it: "I think I'd be too disorganised to get outfits together myself. We do such high-profile things now, like the Oscars. Can you imagine? I'd go mad if I had to do it all by myself."

"It's quite a symbiotic relationship; she's taught me a lot about what suits me and introduced me to other things, because I did spend my whole life in ripped-up biker boots or brogues and socks and a floral dress. I had almost rejected the idea of having a kind of figure, for a while. Almost rejected the idea of things fitting me. Certainly I didn't care whether or not they did. I've never really worried about dressing sssss-sssss-sexily, she finally spits it out, as though it's a rather shameful swear word. "It was more vintage, grungey, pull anything on and throw a cape over it."

She describes herself as a quite chubby and awkward child and a really late bloomer. I didn't start coming into my own until I was about 16." Despite her face gracing the likes of a Vogue cover, as well her distinctive look, people don't approach her often. "Not as much as you'd think. I think people always expect me to be seven foot tall and to be a lot bigger - in energy. But I have a quiet way; I don't feel very starry."

"If people do come up, they just say hello and I don't really mind that. It's like the whole world is one big acquaintance now. Usually everyone is really polite and sweet, which is nice for me because I'm quite shy and it always felt awkward at school, and parties were quite nerve-racking, so now it's almost a level that's been removed and it's just easier to chat to people." The downside of fame is "the privacy thing. Like going to the park and then seeing yourself in the papers two days later - it gives you a bit of a funny feeling. But it's not really my place to complain. Like Kate Moss said, 'Never explain, never complain.'"

Does she ever take a bad photo? "Yeah, I've got a funny face! I can look quite hard if people put too much make-up on me - I can look very stern. My normal face, if I'm not being animated or talking, if I'm just sitting, I look really angry."

She definitely has a quite unmodern look (I'd say she s a beauty, albeit an unconventional one). One can imagine dropping her into the old photographs of Edith Sitwell, Ottoline Morrell, Virginia Woolf and her sister Vanessa Bell et al. "I like the idea of being like Virginia Woolf; in the old photos she looks very beautiful to me. I've always been attracted to the idea of faded, Grey Gardens, mad old lady glamour. Wallis Simpson and Patti Smith are other style heroines and I have a massive girl crush on M.I.A. - her video for Bad Girls is the best music video I've ever seen." She also has a crush on a rapper called A$AP Rocky - "He's got this mouth full of gold teeth and he's really beautiful. He's so cool. You know, Comme des Gar ons? He has this hat that says 'Comme des F*** Down'."

Who would she fancy playing her in the Florence Welch biopic? "If David Bowie in his Ziggy Stardust phase could come back and do it, that would be fun. She's looking forward to a proper break - the record company has promised one - so that she can write her third album, as well, presumably, as do up her new home, and have some fun with the new boyfriend. She's also apprehensive: "Maybe I should keep working in case I get into trouble if I don't! I might get bored and - I don't know - go to too many parties!"

Does she imagine herself doing a Lily Allen and giving up rock 'n' roll for a more normal life? "I think you crave stability when you're in an environment where everything seems transient. It's a nice idea to be settled, and I'd like to get married and kids would be nice," all this said in her dreamy voice, before she snaps to: "But I'd quite like five years before any of that happens."

Florence Welch takes to the stage at Radio 1's Hackney Weekend - airing on Sunday at 8:00pm on BBC3. 

See her in action in the video for Shake It Out below...