The Cocteau Twins had been apart for seven years, the mystique they had attained during their lifetime gradually growing and their influence spreading, when the announcement came that they were to reform. The world was told they would be headlining the 2005 Coachella festival in California, and would follow that with a major tour. According to bassist Simon Raymonde, the band stood to benefit to the tune of £1.5m each for getting back together – enough to guarantee them financial security, enough to secure the future of Raymonde's Bella Union label.

There was just one problem. Within weeks of the announcement, the group's singer, Elizabeth Fraser, announced she wouldn't take part.

"I don't remember it being that much money and in any case that's not the reason [for reforming]," she says today, in her first interview since the band split in 1998. "But people get so fucking carried away. Even though something's staring you in the face, people just cannot see it. I knew it wouldn't happen and it didn't take long to want out."

Fraser's decision to pull out of the reformation was made for the same reasons that contribute to the band's split in the first place: she could no longer face working with the group's guitarist, Robin Guthrie – her lover until 1993, and the father of her first child. But while they were together, the Cocteau Twins established themselves as one of the three main pillars of British alternative music, alongside New Order and the Smiths. Guthrie provided shimmering swathes of effects-laden guitar – surely the inspirtation for what then Radio 1 DJ Steve Wright's spoof rock critic character called "sonic cathedrals of sound" – while real critics swooned over Fraser's otherworldly and often incomprehensible vocals, one describing her singing – to her embarrassment – as "the voice of God." Madonna loved them, Prince wanted to sign them and Scritti Politti's Green Gartside said the vocally gymnastic Fraser was his third favourite melodicist of all time (behind Paul McCartney and Brian Wilson). You still hear their influence in any band striving to sound ethereal and otherworldly. Now her voice is to be heard again, as she releases her first solo single, Moses.

As Fraser tells it, that abortive reunion was not a career move, not a chance to gain the financial reward the Cocteau Twins' reputation merited: it was an attempt at healing. Their childhood friend-turned roadie-turned-manager wanted them back together "so that everyone could be friends".

However, Fraser almost immediately had reservations. "There's still a sense of being committed," she explains of her relationship with Guthrie. "But we're not committed. We're that different from each other now." She describes those differences in a sentence: "You take each other's breath away by doing something or saying something they never saw coming." She now can't even think about her bandmates, never mind see them. "They were my life. And when you're in something that deeply, you have to remove yourself completely."

It's not just her former bandmates she has left behind. For the last 12 years, she has barely engaged with music. She sang on Massive Attack's 1998 album Mezzanine and exquisite hit Teardrop (and toured with them in 2006), but that's it. She's been offered sums "beyond your wildest dreams" to collaborate with other artists – "the weirdest one was Linkin Park" – but all have been turned down.

Fraser sees making music as inseparable from her emotions. She has always struggled to write lyrics, she says, but suddenly something will click and she "goes with the sound and the joy" – that's why she sings sounds and words that have no meaning, of which she can only make sense later. As she puts it, "I can't act. I can't lie."

The inability to pretend is evident even now. She is so nervous before the interview begins, she's actually shaking,

"I live in here," she explains, exasperatedly, pointing at her head. "And it's difficult. I drift with every sensation. At times I'm OK, and at other times I'm such a rubout. My mind just whirrs or stops. There's no middle ground." When she was still performing, she would suffer stagefright. Now she talks of her anxiety spreading to the studio. Her single was recorded some time ago with Damon Reece – Massive Attack's drummer, and her partner of more than a decade – and a close friend, Jake Drake-Brockman. It wouldn't be coming out at all were it not for a tragedy: Drake-Brockman died in September, and Moses is being released as a tribute.

Pop was always Fraser's escape. Back in Grangemouth, a "horrendous" petrochemical town in Stirlingshire, she had seemed destined to follow her mother into the local rag trade until she realised that having "boxing gloves for hands" meant she could barely operate the machines. She would go dancing at a local club, the Nash, which is where she met Guthrie: he spotted the 17-year-old Fraser on the dancefloor one night in 1980 and asked her to join the band he'd started with his friend Will Heggie. By saying yes, Fraser acquired a soulmate and an enabler. "I looked up to him. I could never have done it without him." The distinctive sound they developed "flowed from the chemistry between us", particularly once Raymonde, a Londoner, replaced Heggie.

Fraser didn't feel what they were doing was particularly experimental or original – "It felt like we got away with it," she says – but the early 80s was "a lucky time. Bands could do what they wanted and have a career. It felt like it was feted." But, she says, "It couldn't be sustained."

She and Guthrie were lovers for 13 years, during which time the difficulties any relationship faces were compounded by being in a band together. "We were so close, but certain responsibilities were too much for us," Fraser says. The birth of their daughter Lucy-Belle in 1989 "didn't impact as positively" as she'd hoped.

There were resentments on both sides, she says. They were "outgrowing each other" and Fraser was increasingly unhappy in the band. She resented "doing what people wanted all the time" and began to break free, a process documented on the unusually direct lyrics of the 1993 album Four-Calendar Cafe. The situation was sharpened by Guthrie's dependency on alcohol and drugs, revelations (which came from him, after the band's split) that shocked fans. But Fraser's own unhappiness was unnoticed by her colleagues.

"I turned to others for some sort of reality check, [but] they hadn't even noticed there was a problem," she says. "And that was another thing that sent me absolutely round the bend. When you need things measured and it's not happening it can make you feel quite mad." Fraser endured a nervous breakdown, and underwent a course of psychotherapy. Today, she remains irked by the suggestion Guthrie made after completing rehab, that he'd needed the drugs to make the music.

"I don't believe that and I don't think he believed that in the beginning," she insists. "I mean, I tried to keep up, but I find it difficult enough to communicate anyway. On drugs I just shut down. I just thought they'd get fed up with it, and get into something … healthy." She allows herself a chuckle. "But it never worked out."

Fraser is still furious with herself for staying in the band for two years (and another album, 1996's Milk and Kisses) after her relationship with Guthrie ended – "but we were terribly over-contracted and I wasn't strong enough to stop it". The resulting tensions, she says, caused the greatest lasting damage. "Periodically, my mind is blown," she says, "and I'm swamped in feelings I can't deny."

After parting from Guthrie, but still in the same band, Fraser struck up an intense relationship with Jeff Buckley after they became infatuated with each other's voices. Again, emotion produced music. A sublime duet they recorded called All Flowers in Time Bend Towards the Sun is floating around the internet, to her irritation.

"Why do people have to hear everything?" she complains. I tell her it's wonderful. "But it's unfinished, you see. I don't want it to be heard." There's a pause. "Maybe I won't always think that."

Buckley died in 1997, by which time they had lost touch – Fraser had grown frustrated with his constant touring, a reaction that weighs heavily on her. "I just wish I'd been more of a friend," she says, softly. "His career was everything to him, and I wish I had been more understanding – happy with a different kind of relationship. I missed out on something there, and it was my fault."

The news that Buckley had disappeared – he drowned, swimming in the Wolf river in Memphis – came while Fraser was recording Teardrop with Massive Attack. "That was so weird," she says. "I'd got letters out and I was thinking about him. That song's kind of about him – that's how it feels to me anyway." It seems she is haunted by guilt: for not being there for Buckley, for everything. As she puts it: "I need to forgive myself."

She changes the subject, suggesting we call Reece – touring in snowbound Bratislava – and immediately brightens. Even over speakerphone, their chemistry is obvious.

"Did people tell you to keep away from me, that I was fucking trouble?" she asks him.

He chuckles. "Only one or two."

The couple met shortly before the Cocteau Twins split. He literally whisked her off on his Triumph motorcycle, a memory which gets him talking about fellow motorcyclist Drake-Brockman. When Reece and Fraser moved to Bristol, Drake-Brockman helped fix up the house, and taught Lily to swim. Reece is still in shock after the accident that killed his friend. "He wore a tweed jacket and rode a 1938 machine," he says. "He was going to be the old man on the motorcycle when he was 80."

Talk turns to Fraser, who'd earlier explained that another reason for her withdrawal from public life was having a second child. After the birth of Lucy-Belle, she'd kept on working, but she made sure she was able to watch her second child grow up. "I was so angry with myself," she sighs. "I realised what I'd missed out on. But that's me, angry, angry!"

However, the scars of the past are healing. Despite the burdens she carries she is happy, and now her closeness to Reece has produced her first new music for more than a decade. The couple have even made "what might become an album", music that – untypically – she says she is very proud of. But she insists that – just like the duet with Buckley – the music isn't finished, and isn't ready for public consumption. "I'm very perfectionist," she explains. "I'm getting stronger as a person, but sometimes I just need to" – she's laughing now – "get over myself!"

Reece understands that the process of putting her back together as a singer is an ongoing process.

"I feel sorry for the general public because I hear her singing in the house and it's truly amazing," he says. "But she's absolutely genuine in every way possible. Which can be very frustrating, but is an amazing attribute to have. I've worked with many singers, and a lot of them are fake. The world is a sadder place without Elizabeth singing."

Fraser smiles sweetly, and doesn't say a word.

Moses is out now on 12in and download through Rough Trade