While Tim Burton and Johnny Depp are widely considered to be joined at the hairdo – an emo-hipster version of Scorsese and De Niro – it's composer Danny Elfman who has been the director's most loyal and enduring collaborator. Think of any of Burton's deranged goth-com fantasies, from Beetlejuice and Batman to Sleepy Hollow and Frankenweenie, and it will be Elfman's barbed and bouncy compositions that spring to mind as readily as man-child misfit heroes or morbid, arachnophile art design.

Elfman, now 60, was the frontman of performance art outfit turned rock band Oingo Boingo when Burton asked him to score his 1985 debut film, the comic road movie Pee Wee's Big Adventure. Only twice since then has Elfman surrendered the baton on a Tim Burton film – most recently on Sweeney Todd, which was wall-to-wall Sondheim, and back in 1994 when the pair quarrelled prior to Burton making Ed Wood. "In a quarter-century career, it's inevitable that two people like us would have a massive implosion," reflects Elfman chirpily, remembering how the tension of working on back-to-back films and the pressure surrounding the second Batman film caused him to snap. "We used to joke that we'd end up like Hitchcock and Bernard Herrmann, who had a famous falling-out and then never spoke to one another again in their lives. I was very temperamental back then. And I regretted it soon after. It was like losing a sibling."

Now their work is being celebrated in a BBC concert: Danny Elfman's Music from the Films of Tim Burton, which will premiere at the Royal Albert Hall in London next week before several dates in the UK and US. Burton has contributed visual materials for the event, while Elfman, who has marshalled each score into its own suite, will play on stage. More remarkably, he will sing compositions he wrote and recorded in 1993 for The Nightmare Before Christmas. It will be the songs' first live performance (not counting the millions of children who howl along to them in front of the TV), and the first time Elfman has sung on stage in 18 years.

"I'm scared shitless," he says, raising a weak smile. To be perfectly honest, he looks it. He's a slight, softly-spoken man with orange-tinted specs that match his marmalade hair. We are seated on sofas in a cavernous, wood-floored room in his Los Angeles base, Studio Della Morte, where instruments (several gongs, a discarded accordion on the floor) compete for space with macabre props (cow skulls, dolls in various states of metamorphosis or dismemberment) and oddball paintings (a hare with boxing gloves). "Until I get to rehearsals this Saturday with the BBC Orchestra, I do not know what to expect. It would be typical for me to have bitten off more than I can chew. If I hold true to form I'll have gone a bit too far."

And once he starts talking about singing live, he turns paler than Edward Scissorhands. "I don't think things through too well," he grumbles. "Why don't I say: 'Hang on: I've never sung these songs live – or even all the way through'? Me being me, I say, 'Sure! Why not?' Then I realise, 'Holy crap: these are harder to sing than I thought.' They're very aggressive songs, at the top of my range, lots of shouting, very theatrical. And I haven't sung in so long." Why not? He narrows his eyes comically. "I've been a little bit busy scoring, ooh, I don't know, 80 or 90 films … "

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Fifteen of those have been written for Burton (they start work soon on their 16th, Big Eyes, a biopic of the artist Walter Keane). "People expect us to have some invisible shorthand," Elfman says. "But it never gets any easier. Tim is just as complex now, maybe more so. Every time I play music for him I'm as nervous as I was that first occasion. Countless times, I've played something and watched him just put his head in his hands and start pulling at his hair. With Tim, I've never been unhappy with where we've ended up, but most of the time we've had to spiral around quite a bit to get there." Working with any film-maker, he says, involves a degree of telepathy. "Writing the score is the easy part. Getting into the director's head and understanding their psyche is what's hard. But that's what you need to do. You have to be half-composer, half-psychiatrist."

For the concert, Elfman has written new material to act as connective tissue – and, I would wager, to keep himself interested. He admits that the past holds no fascination for him. "Once it's done, it's done," he sniffs. "I never look back. I never get nostalgic. That's just the way I'm wired. Remember the guy in Memento? That's me. 'Oh yeah, I guess I used to sing, because I have tattoos that say I did … '" At that moment, I catch sight of his bare calf: the skin is indeed crawling with gecko skeletons, patterns and symbols, all dating back to his Oingo Boingo days. He hoists one sleeve to show me some more unfinished tattoos on his left arm. "These were done 25 years ago. It was supposed to be a whole arm and leg, but I never got them finished."

Stretching as they do almost all the way back to the date of his first score for Burton, there seems something symbolic about this body art: a testament to the way his life was going before he was tempted away from rock music. Elfman learned on the hop; for five years or so, he regarded scoring as a hobby, with rock'n'roll his office job. His association with Burton was not always embraced by others: composing the score for Batman was "the fight of my life and my career. The studio didn't want me. I had no experience doing that kind of movie. The natural tendency is for them to say, 'Can we please get John Williams? Or whoever can do a John Williams.' Clearly that wasn't me. I really had to prove myself." All the while he was looking for a way out of rock music, hobbled by the guilt of abandoning his bandmates. Finally it was his health that forced him to give Oingo Boingo the bounce. Years of standing on stage every night in front of monitors emitting screeching feedback had given him hearing loss and tinnitus, from which he continues to suffer today.

There's one aspect of his former life he misses: "The sweat," he sighs wistfully. "In the middle of writing a score, I'm so happy to be creating rather than playing the same songs night after night. I hate repetition. But I don't have the joy of pulling off a T-shirt, wringing out the sweat and throwing it at some guy in the audience. I used to love jumping into the mosh pit, then climbing back on stage with red claw marks all over my body." I suggest he could give it a go at the Albert Hall, but he doesn't seem convinced.

Danny Elfman (left) on stage with Oingo Boingo in 1982 Photograph: ABC/Getty

It's appropriate that Elfman will be performing songs from The Nightmare Before Christmas: not only has he gone on record as saying that it is his favourite of his works for Burton, but it's also the one with the deepest autobiographical roots. The friction between Halloween Town, ruled over by Jack Skellington, and the far cosier Christmas Town, which lies outside Jack's dominion and comprehension, is for Elfman symbolic on several levels. He equates Jack's struggle against the confines of the familiar with his own discomfort as a rock musician. (Even as he was scoring the movie, he was trying to extricate himself from Oingo Boingo: "I felt like I was writing about me and my band," he says.) He also sees in the conflict between Halloween and Christmas a striking replay of his own childhood tensions. "Halloween was always my favourite night of the year, and Christmas was the saddest. I was raised Jewish in a secular family. We didn't celebrate anything, so in my mind all my friends were singing Christmas carols in a warm, happy environment while I was stuck in this depression. Black clouds gathered. I was cast out. Halloween was the opposite. It was the night to become something else. Anything! The night to go wild. We were boys loose in the world, running amok."

He tells me that while composing the songs for The Nightmare Before Christmas he would run them past his daughter, Molly, who was eight or nine at the time, to get her approval. Now when he's scoring a children's film (such as this year's Epic or the forthcoming DreamWorks animation Mr Peabody & Sherman), he keeps in mind his youngest child, eight-year-old Oliver, whose mother is Elfman's wife, the actor Bridget Fonda. Elfman and Fonda met on the set of A Simple Plan, which he was scoring, and in which she was starring as a woman whose husband finds a stash of abandoned loot in the wreckage of a crashed plane.

Composers and cast members typically occupy different worlds – Burton is unusual in inviting Elfman to the set for a day or two on each shoot ("I might just say hello to Johnny … "). But the director Sam Raimi, who made A Simple Plan, had also wanted Elfman to come on set to get a taste of the mood. Fonda and Elfman said hello, then didn't run into one another again until a party six years later. The connection was enough, though, that it enabled them to strike up a conversation. And the rest is history.

Not surprisingly, Elfman seems awfully fond of his score for that unsettling movie. "Sam gave me lots of freedom to do something odd, which is where I'm happiest. Out-of-tune pianos, detuned banjos, chords where one part goes sharp while the other goes flat. I had a great time." He smiles. "And Oliver can thank that film for his existence." I ask what piece of his father's music Oliver is fondest of, expecting it to be the theme from The Simpsons, which Elfman conjured up in a morning. "No, he's never watched The Simpsons. Maybe Spider-Man or Men in Black." It transpires that he is more likely to notice the ones his father didn't write. "Sometimes he gets cross with me. 'You did this one, right?' 'No, I actually didn't.'" Elfman imitates the disappointed groan of a disgruntled child. "'Why didn't you do Avengers Assemble?' 'I'm sorry, son.'"

• Danny Elfman's Music from the Films of Tim Burton is at Royal Albert Hall, London (7 Oct); First Direct Arena, Leeds (8 Oct); SSE Hydro, Glasgow (9 Oct); and NIA, Birmingham (10 Oct). Details: apolloliveevents.com