"There is no actual milk," says Björk, surveying her fridge. It is a common scenario, simultaneously being played out in millions of homes worldwide, but chez Björk even the prosaic feels surreal. We have coconut milk in our tea instead. "If you don't mind how it looks, it's OK," she reasons.

She's just so… Björkish. First, there is her look. Today, a dress made of geometric panels in off-white with an exaggerated puffed sleeve. Later, when we venture out, she pops on a long, padded coat covered in metallic swirls – how Barbour might have interpreted Joseph's dream coat. Then there's that unmistakable speaking voice, a clash of trilled Icelandic "rrrs" and a cockney chirrup acquired when the "drrrrrumer" of the band she was in got them gigs in the UK and they stayed in squats. "English for me is still like an arm's length removed," she says. "You are always a bit different in the mother tongue. That's why it's maybe easier for me in English to be an extrovert. In Icelandic I'm more private."

A woman in a long, felt tunic, no feet visible, wearing dark glasses and a straw hat glides into the room. "Oh," says Björk, "this is my mum." Björk's mum, Hildur Rúna Hauksdóttir, is excellent company. While her global-superstar daughter valiantly attempts to rustle up lunch (as she's only on a flying visit home, Björk hasn't been to the shops so is low on all provisions – not just milk), Björk's mum smokes outside and tells me conspiratorially that a distant ancestor was an Irish girl enslaved by Vikings. She is worried about the tourist influx to Iceland, especially the enormous cruise ships and what their impact might be on the natural world. Later, I read that she went on hunger strike in 2002 to protest against the building of a power plant in the Icelandic wilderness. Her daughter appears with a salad, some wooden plates, one of which has warped in the dishwasher, and some rice crackers. We eat lunch in the front garden.

It's all pleasingly bohemian – a bit how I imagine the hippy commune Björk famously grew up in might have been (although she says this has been overcooked: "It was a group of kind people with long hair, who lived together and listened to Jimi Hendrix, but they did have 'normal' jobs.") But given her international stardom, I had expected more retinue, fanfare, housekeepers and less eating in full view of the passing public. "Oh, we're not really like that in Iceland," Björk explains. "It's a place where if you go to the geothermal baths the prime minister is naked in the shower. There is no hierarchy. In every family there will be one poet and one bricklayer, and there will be, you know, a sculptor and it all merges into one thing."

The same egalitarian principles cover Icelandic life as a whole, according to Björk. "There are no wars between the genres and wars between the classes and there are no wars between people and the arts."

At this time of year, January to June, Björk is usually to be found at her other home in New York's Brooklyn Heights, where her daughter, Isadora (with artist Matthew Barney), attends the liberal arts school, Saint Ann's. It seems a very different life from this one here in Iceland, where she's clearly invigorated by the natural environment. But she talks enthusiastically about Brooklyn Bridge Park. "When it's ready it'll be almost as long as Central Park," she tells me. "Plus, it's got the East River on the side, so you get the wind from the Atlantic Ocean." To miss the wind seems very Icelandic.

Björk on tour with her Biophilia album and project. Photograph: PR

Björk is back in her homeland working on her Biophilia project, which is billed as "a multimedia exploration of the universe". First released as the 2010 album of the same name, it is a giant collaboration between app developers, scientists, instrument makers and a 24-women Icelandic choir who toured with Björk for more than three years in a stage show. Later this year, a film of the live show will be released, narrated by David Attenborough, who is a huge fan of the project.

The great biologist Edward O Wilson coined the term "biophilia" in his 1984 book of the same name, hypothesising that we are hard-wired to associate with other forms of life. You can interpret it, too, as an impulse towards environmentalism and conservation: we need nature and therefore we need to protect it. Björk's interpretation is more playful. What began as simply an artistic concept – "You know, just picking 10 songs with 10 emotions and 10 connections to the natural world" – became a pioneering "app album", whose iPad form gave each track an app of its own, incorporating interactives and even games with the music. Now it has evolved into a way of teaching music and science using patterns from nature.

Björk is fresh from three days of meetings with Scandinavian educationalists because in several countries, including Iceland, the Biophilia Educational Program is soon to be on the curriculum (or "corricoooloom" as she pronounces it) funded by the Nordic Council. It is hard to imagine Michael Gove signing UK schools up any time soon, but, as Björk puts it: "Education is one thing we're good at." She says she herself studied musicology and classical notation, but the system never encouraged her to actually form her own songs. "Basically music schools are just like conveyor belts for violinists and cellists for symphony orchestras," says Björk. "There is not usually any space for students to have a voice. The average person on the street is made to feel they can't join." Biophilia's programme, designed to be hands-on and non-academic, has been "really popular with kids who have ADD or dyslexia". She sighs. "Unfortunately it means we have to sit down and write a curriculum and that's a contradiction."

Her native Iceland is a biophiliac's paradise. Not 30 minutes from Björk's house you can watch the tectonic plates shift (well, sort of… they move an inch a year) at Thingvellir – track nine on Biophilia, "Mutual Core", deals with plate tectonics. Pretty much anywhere in the country you can watch waterfalls plunging into the ground or trek across lava fields. The Icelandic historian Guðmundur Hálfdanarson has suggested that the natural world is so inextricably entwined with his countrymen's psyche that "nature is close to replacing language as a symbol of Icelandicness".

Late in the game, the rest of the world has caught on and Iceland has bounced back from its cataclysmic banking collapse of 2008 partly thanks to tourism. Last year there were 825,000 visitors to Iceland – that's 2.5 visitors per resident. Tourism now exceeds fishing as Iceland's dominant industry. Björk has helped put this country on the map, but so too did the unpronouncable volcano Eyjafjallajökull: bookings surged after the ash cloud spread south. The news coverage seems to have served as a tourist information video as we collectively swooned over Iceland's theatrical natural world. Game of Thrones may have helped, too.

So it is a tragic irony that as Iceland grasps how to exploit its volcanic tundras for the biophiliac in all of us, it faces its greatest environmental threat. Activists accuse the government of industrialising this great pristine wilderness, particularly the highlands, through back-door deals with foreign aluminium giants involving tax incentives and undeclared sweeteners. Iceland's geothermal energy is a cheap way to refine bauxite mined in Australia and the US into aluminium, and without using fossil fuels. But don't be fooled that this is a green, impact-free way of doing industry. Geothermal requires the diversion of major glacial lakes through massive damming projects that will change Iceland for ever, while the effects and exploitation of geothermal fields has been shown to cause irreversible habitat damage through pollution and emissions.

In May last year, the global aluminium trade got a boost: the left-leaning government was ousted by a centre-right coalition apparently hell-bent on opening up the highlands to industry in the name of economic development. It took the new government precisely one day to begin pushing ahead with a stalled new aluminium smelter in Helguvík.

Björk has entered the battle. "I want the campaign to get foreign interest," she says, carefully. "It can't only matter to us – it needs to matter to everybody to get it stopped." She wants everyone to ask themselves the question: do you want to open up aluminium production for a handful of corporations or to preserve one of Europe's last remaining pristine wildernesses? "Otherwise the rednecks [as she habitually refers to the pro-aluminium lobby] could build 17 dams and then it's all over, even though the majority of people here don't agree with them."

A homegrown environmental movement is relatively new for Iceland. Previous campaigns tend to have been carried out by the big international NGOs, such as Greenpeace and Friends of the Earth, on fishing and whaling, where Iceland has often been cast as the bad guy. The cultural touchstone for this nascent Reykjavik-based movement is Andri Snær Magnason's 2006 book (and later documentary), Dreamland: a Self-help Manual for a Frightened Nation, which explores the damage of foreign ownership and predicts the privatisation and industrialisation of the wilderness.

Inevitably, the "rednecks" paint the opposition as a metropolitan elite who don't have to concern themselves with the economic realities of the deprivation in the highlands of Iceland, where the fishing industry has collapsed. Doesn't she get flak for being a part timer, spending half the year away? "I do sometimes have to answer those questions, yes," she says. "But even venture capitalists here now understand that our future needs to be in nature, not destroying it. There are bankers who now have companies taking boat trips across the fjords. That's what's turning people around.

"I'm not saying we go back to the past and live in a cave, but to have a smooth route into the 21st century it makes sense to embrace technology and give ourselves options, not have a dirty industrial revolution. We need to take a short cut to the green shit!"

How far would she go to protect the wilderness? It's not hard to imagine the fierce, charismatic Björk we see on stage staring down the police to block a dam project. "I don't want to judge direct action," she says quietly, "but it's a bit masochistic, right? What would chaining myself to a bulldozer achieve?" She has a far more pragmatic approach. "I remember going as a 16-year-old to England and everyone was vegetarian because of their principles and we just didn't get that here," she explains "We can't grow vegetables so well, so it's a different context, you know? We're functional about things."

Instead of manning the barricades, Björk turned to the context she knows best and held a concert. With various luminaries on the bill, including Patti Smith and Of Monsters and Men, Björk raised 35m króna (£3m) for the campaign. "That's a lot of money in Iceland," she adds. "Money talks, bullshit walks."

And what will they do with the money? "We decided we're going to start a national park in the centre of the island. Instead of fighting with the rednecks, we'll just get on and do it."

That's the thing about Björk, she's incredibly driven – much more than the dreamy, raised-in-a-commune hippy mythology would have you believe. Without this inner steel, it's hard to imagine how she would have become the cultural force she is. In the 90s she was as big as either Gallagher brother, pushing the boundaries of electronica and house music when previously in dance music women were mainly used to supply anonymous (and often trite) vocals over the beats.

She also navigated fame with a baby – at 20 she gave birth to her son, Sindri (now 27). "On my first tour with the Sugarcubes with him I was like, 'OK, this kid is really going to love being on the road or hate it. If he hates it I'm going to have to go back and work in a fish factory.'" History records that the fish factory wasn't necessary, but it must have been tough. "I guess I was lucky. In Iceland everybody at that time had kids young. You know, we win all those surveys that we are the best place for women. So I didn't have to worry about a lot of stuff that was probably more complicated for women in England."

Although she made her choices very young, she made pretty good ones, often by instinct. Despite offers from Warner, Sony and other music industry giants who came to Iceland to try to sign the Sugarcubes, it wasn't Björk and her friends' philosophy to sign big. Instead, her friend Derek Birkett, a bass player in punk band Flux of Pink Indians, started the record company One Little Indian around them at the indie record shop where they all worked shifts.. "These were not the type of people who say: 'Leave your kids at home when you tour,'" Björk says.

Nearly 30 years on, Birkett is still her manager and One Little Indian her label. They still own a record shop, Bad Taste, in downtown Reykjavík, and she says they've never had a formal meeting. "We just sort of go for a walk and talk about our kids and then he goes: 'Do you wanna play Tokyo next year?' and then we go from there." Their enduring relationship is one of the keys to Planet Björk. "The pressure only comes from people who want you to repeat yourself and we have shaken off those people," she says. "The only ones left are the ones who understand I have to do things how I do things."

Birkett is the sole publisher of all her work, from the tiny, idiosyncratic books to the massive, swirling projects like Biophilia. This is critical to her because it allows her to create more. "More people get burned out by not putting out enough," she says. "Once you've done something you need to get it out there otherwise you're all stifled."

After lunch, Björk's mum drives us to the local cinema to watch a screening of Biophilia Live. The film has been remastered and Björk wants to check the sound and see what some of her friends think. Her form appears on screen in a huge ginger wig, a blue chinstrap extravagantly painted on her face and a dress like some intergalactic mollusc. In her seat she listens intently.

There are not many artists who can combine the lifecycle of a jellyfish with a breakbeat and make it work. But this is an extraordinary piece, perhaps more an opera, where Björk and drummer Manu Delago are at their virtuosic best. It's utterly bonkers yet moving – especially a strange love song set to a mutating virus. It should be mandatory viewing for anyone about to dam a glacial river to facilitate aluminium smelting.

Afterwards, Björk seems happy. Her extended clan of family and friends stands outside chatting, enjoying the Reykjavík wind. "That man is Björk's father," Björk's mum nods towards a neat man in a sports jacket. These days he looks neither like a commune dweller nor even a Hendrix fan.

Before we part, I ask Björk if she could ever see herself committing in a more permanent way to the environment, perhaps in a political sense. She tells me a cautionary tale. Her comedian friend, Jón Gnarr, who began a political party for a joke, got elected and now, as the Mayor of Reykjavík, has 14 hours of meetings a day. "At the moment I get to write songs as much as I want," she says. "Sometimes I go to meetings and that's important, but they do drive me kind of nuts. I would not want to increase the meetings or decrease the music making, you know what I mean?"