This is a story that almost didn't happen. And I am on a train to Devon to meet the people who were determined to see that it did. The saying "Be careful what you wish for", in the case of the Spicer family, needs adjusting to "Be careful what your brother wishes for". It is the story of three siblings: Kate, a 42-year-old London-based journalist, her youngest brother, Will, a 36-year-old film-maker, and their middle brother, 40-year-old Tom, born with fragile X syndrome, the commonest cause of inherited learning disability, affecting about one in 4,000 men and one in 6,000 women – a sort of autism, caused by a mutated gene on the X chromosome that can inhibit intellectual development. For half a lifetime, Tom had been a fan of heavy metal – Metallica his favourite band. And, by 2009, no one was in doubt about Tom's dearest wish: "I wanna meet Lars." Lars Ulrich is Metallica's drummer. Many times a day, Tom would repeat his wish. It is like this with fragile X sufferers: an idea jams – and language gets stuck with it. The technical name for this is "verbal perseveration".

Will and Kate had talked before about making a film with Tom (he would "map read", they'd concoct a random road movie). But now, the Lars fixation got them thinking. Could they make Tom's dream come true – and get him together with Lars? "We began to think we could exploit our media positions to do something genuinely cool for our brother," says Kate. Put like that, it sounds easy. But, as they were about to discover, "ease" was not to be part of the process. What they have produced is a film that will make everyone who sees it want to champion it. It is original, funny and overwhelming – and it will make you cry. It is called Mission to Lars.

One can think of several Oscar-nominated films about autism and other disabilities – Rain Man, Ryan's Daughter, I am Sam, Precious – and, more recently, Extremely Loud and Incredibly Close – but they can be counted on the fingers of one hand. There are 1.5 million people with a learning disability in the UK – and they are feebly represented in the arts. This truthful, unglamorous film with a learning disabled, middle-aged man as its centre is not only a rarity, it is a treasure.

As the Devon landscape rushes past, I picture the Spicers setting out in the opposite direction, to LA and beyond – and of the moment when Kate and Will ask their parents for tips about Tom. They are given a lengthy list – it reads like a warning. Tom is dependent on routine but, on their transatlantic trip, uncertainty will rule supreme. There is no guarantee Tom will even meet Lars. Only one thing is sure: wherever Metallica goes, the Spicers will follow. We watch with horror as they rent a mobile home and drive this overblown beast along the bumpy concrete of interstate 15 to Las Vegas. Tempers fray. And then comes the show-stopper. Tom has cold feet. He does not want to go to a Metallica gig. But doesn't he want to meet Lars? "No." It looks as if dream and film are over. The trip has become a nightmare. Kate and Will are catatonic. Head in hands, Kate admits to her brother: "I don't think you and me are very good at looking after Tom…"

But as the audience we know that if the mission were not accomplished, there could be no film. And, against the odds, we are eventually backstage, chez Metallica, where Tom and Kate endure a heck of a wait. Kate gnaws at her nail varnish, Tom is restless. But then the door opens and a small, ordinary Danish man in a white plastic bomber jacket walks in (the truth is that when Lars first moved to LA, he was a junior tennis champion before founding Metallica). He is smiling. And by this stage, anyone watching will be crying hard – because Lars is so nice to Tom. He is not patronising. He is sensitive. On the back of this film, Lars is about to become everyone's hero. And the look on Tom's face is – but you have to see the film to witness it – happiness beyond words.

Bystock Court is a residential care home on the outskirts of Exmouth, where Tom has lived for 22 years. I travel there with Kate. She is funny, hyper-intelligent, attractive and anxiously controlling. She thinks of everyone and everything (she has engagingly instructed me to bring "pocket money" for Bystock's excellent eggs). But you can see she is making efforts not to take charge. Will drives into the car park in his Volvo – a family man with three children. He has a video production company. He is not as dissimilar from Kate as each of them likes to think. He is personable, nice, amused – the smile, usually missing from the film, is lovely to see. But he is more self-contained than Kate – on the back foot.

The house itself is double bow-fronted, red brick – Victorian or earlier – with a generous garden. It has a comfortable, battered grace. And with summer making its surprise appearance, residents are coming and going, talking to each other and, sometimes, to themselves. I sit on its elegant terrace. It is the only time in my career – and it is refreshing – that my first sighting of the person I have come to interview is his disappearing back view. Tom, instantly recognisable from the film – owlish glasses, sweet face, loping gait – hurries past with a mischievous smile. He is looking at his feet critically and says (his speech is not always easy to follow), "It is a mess", before heading in the direction of the shrubbery.

It is not clear what is a "mess". But one reflects, with amusement, that he is right – if what he is talking about is family. Kate tells me later that Tom was just winding her up. But it reminds me of the first crisis in the film when Kate and Will fail to show up on time to take him to America and Tom does a runner. And they had already been through so much to get the film off the ground. Kate fills me in on the hell of fundraising, unenthusiastic television producers ("It was a big, fat no") and their unflattering insistence that she lacked the "charisma" to be a presenter.

When a television company finally came good, the Spicers decided against it. They no longer wanted to "entrust the family dynamic" to anyone other than their long-suffering co-director, James Moore. It was John Battsek, of Passion Pictures (responsible for Man on Wire), who inspired them to go it alone. He said: "Sod the telly, make the film you want to make." They were determined to raise awareness: "There is still so much prejudice where it is not supposed to exist." And Mission to Lars now has charity status. The film's fundraising has earned Mencap £25,000. And the relationship with Mencap will continue. There are plans to reform Mencap's leisure provision to film an annual "Mencap mission" in which other learning disabled people can pursue their dreams.

But there is one score that needs settling: why was it so difficult to reach Lars? Couldn't Kate, as a journalist, hook him with a charming phone call? She explains it was not simple: Metallica are the biggest metal band on the planet (with 25m likes on Facebook) and wrapped up in red tape. "I don't know what did it in the end – whether it was the correct emails – climbing the greasy pole to the top – or when I was drunk at a party late at night and met someone who surfs with Kirk [lead guitarist Kirk Hammett]."

And besides, in the film, the axis of worry shifts. It is not about whether Lars will see Tom. The new question is: will Tom refuse to see Lars? "It was Tom who was behaving like the rock star," Kate laughs. But his reaction, typical of his syndrome, is sympathetically presented. It comes across as a larger-than-life version of what all of us feel when something we long for is suddenly in the offing: stage fright without a stage. What is amazing – and partly explains the film's power – is the Spicers' emotional honesty. Rough and unready, they don't finesse anything. Kate openly calls her family "dysfunctional".

But to talk about her parents, she says, would be "to open a can of worms, kick over a hornet's nest and uncover a viper's nest". This does not stop her or Will returning to the subject. Their father – a surgeon – and mother divorced when they were little (Will was a baby, Tom four, Kate six). They now hazard that the stress of having a child with fragile X must have been "a big factor" in the marriage breakdown.

The family split in two: Will lived with his mother, Kate and Tom with their father. But it is their stepmother, Jane, who is the most intriguing parent in the film, the only person with authority. Kate's mother, admirably, makes no secret of the fact that Jane is more effective with Tom than she is. Will says he only now realises (after having a bash at Tom maintenance) how hard it must have been bringing Tom up. Jane had three more children of her own – so was often in charge of six. He used to see her as "slightly cruel". Now he sees that what is implied to have been a dictatorial rule was expedient. His mother he sees as a "complete softie".

Will explains that one reason he wanted to make the film is that he has "struggled" to explain Uncle Tom to his children. He hopes the film might do the job in a "light-hearted and cool" way. Kate, who at present has no children of her own, feels slightly differently and is less lenient about the past. "This is so not a film about my parents. I love them to bits, but they won't be there for ever. Tom will be my responsibility. I will have to fight for his wants and desires in a world that, at best, cannot afford to fulfil them and, at worst, doesn't listen or care.'

Tom was 11 when he was diagnosed with fragile X. Kate remembers from childhood "the strain in people's voices. They'd be saying he was not normal, potty-trained, crawling. You absorbed the adult anxiety – and euphemisms." Tom's condition was not much discussed: "We don't talk much as a family." Tom would be referred to as "mentally retarded". It was "awful". Diagnosis, we agree, is a difficult subject. Kate thinks there is an over-diagnosis of autism in this country and the US. We consider the attendant risk that a person will be defined by a diagnosis. "You have to ask: at what point do we stop diagnosing and say: This is Tom." Kate then tells me about her boyfriend's joking reaction to the film – which has its serious side. He asked: "Which one of you is supposed to be normal?"

I ask Will whether he thinks that when one family member has a disability, it is, in a sense, shared by the other children? "That is true. But you don't question it. It is your life." It was not until Will's wife was expecting their first baby that they woke up to what fragile X, a hereditary condition, might mean and were horrified by their ignorance. Kate says: "It galvanised us. Until then, we had existed apart from the generation above. Fragile X was their problem."

They made it their business to learn about fragile X. "It even affects fruit flies – it goes back to the primordial soup. And now we can see," Kate adds, "there were lots of Toms in our family tree. We look at black-and-white family photos and say: 'We think she was a Tom.'" They also got themselves tested (and aren't carriers). But Kate admits: "I have often wondered if fragile X made me internally anxious about settling down and having a child."

Kate is ashamed to remember herself as a child. She was often Tom's tormentor. Once, aged 10 and driven to distraction, she ran to the kitchen and "grabbed one of Dad's carving knives from the magnetic rack and held it to my brother's neck: 'You will do what you're told or I will cut your throat'." All through his childhood, Tom had "humdinger tantrums that Mum now, knowing more, thinks were small fits: petit mal seizures". Today, she is fiercely sisterly towards Tom and would be more likely to brandish the carving knife at anyone who treated him badly (boyfriends who could not be natural with him never lasted).

"Tom can be upset if he is in a room of people and feels left out. He can be dismissive of other residents. He will tap his head and say: 'They have got issues.' He is unfathomable: he can surprise you or be obtuse. But he is not stupid. If you are calm with him, you can have good chats."

As teenagers and when they were in their 20s, there was one place where they were together and happy: in Will's old banger – commuting between their parents in Bristol and Devon in the school holidays. "We'd eat junk food and listen to AC/DC tapes." It was "magic" – old banger heaven. And that is what they were trying to recapture in the film. "I always expect things to be easy – and they never are," says Will. His starting point is: "We are brothers and sisters, we have never not got along." But he adds: "Kate can be hard work." And by the end of filming: "We were raw and ragged. It felt wrong… And we didn't want the film to be about film-making – that was a constant tension."

Kate continues: "There were so many fights. Will would make me feel like a loser. Tom, responding to our tension, would look distraught and unhappy. I felt so lonely with nowhere to run." Now she says: "We were like three little islands." How often did they think of giving up? "It was non-negotiable. We couldn't let Tom down. The film took on a life of its own. It was a monster that had to be fed."

By the end of the trip, the Spicers were not on speaking terms. When they got back to Bystock, a bottle of red wine, badly packed by Kate, had broken in Tom's case. She and Will had a "screaming fight" as a parting shot. "We were totally out of love with each other," Kate says. Even by the cutting stage, they were still asking: are we going to make this work?

Once the film had been made, there was the further problem of Lars. There was no telling how he would react. Metallica, who formed in 1981, are one of thrash metal's "big four", alongside Slayer, Megadeth and Anthrax. But their members have not always been model human beings (as Joe Berlinger's 2004 documentary Some Kind of Monster revealed; while the 2000 lawsuit they brought against Napster helped effect the file-sharing site's demise).

And their stage image does nothing to reassure. Kate tells me about the gig she and Tom went to – part of the World Magnetic tour, in Anaheim, home of Disneyland: black helium balloons, with Metallica insignia on them, a revolving drum kit for Lars, coffins lowered onto the stage mid-act. It seemed possible that the film itself might be buried alive.

Its fate was decided last June at a small screening at Alfred's club in Mayfair. All the Spicer family assembled for the occasion. Lars arrived really late – and Kate had a second chance to strip her nails of varnish. But Lars turned out to be "really cool". Kate sees it like this: "When artists get to that level, they have either to evolve as human beings or become the biggest knob. Lars tries hard to be authentic. I think that is because he is Danish." And at the end of the film, Lars "jumped up and down and cheered".

Tom does not linger in the shrubbery – and turns out to enjoy having his picture taken. We drink tea together. You're famous, we say. "Yes, I know," Tom laughs. He is pleased with "my film" and the promotional poster and leads us to his room to find space for it. A togetherness is visible now between the Spicers – which was what they all wanted "deep down".

Will and Kate are seeing the benefits of having come through it. And so is Tom who has, since the trip, become more independent: taking bus journeys on his own, using a mobile, visiting London. Tom tells us, approvingly, the film will be loud. "But you don't like that?" Kate exclaims (hypersensitivity is one of the causes of the intense social anxiety that is part of the fragile X syndrome). "I do now," Tom says. The world's foremost expert, Prof Randi Hagerman, explains in the film: "Tom hears the world 10 times louder than anyone else," but he seems to have a love-hate relationship with noise. Today, "really loud" is his catchphrase. The key is to control the level – in the film, headphones save the day.

Tom announces that his Metallica T-shirts are "in the wash". Kate ignores this. She opens a drawer stuffed full of Metallica shirts. Tom shows me the drumsticks Lars gave him and assorted DVDs and pictures of the band. He appears to be developing a crush on James Hetfield, the lead vocalist (famous for accidentally setting fire to himself at a 90s gig). He tells me about Hetfield's superior size. (He gravitates towards strong male characters.) I ask how big Lars is? "He's tiny."

But there is no cause for serious concern: the devotion to Lars is holding. Tom thinks it is Lars's turn to visit him. His eyes shine as he imagines his friend giving a gig on the lawn. Kate thinks a reality check is needed: "Lars would never come." Yet somehow the fantasy builds. We talk about what Lars would eat: "Mum's cottage pie." And he would drink "tea and wine", Tom adds, and "more wine". "Chardonnay," says Kate. Tom, she tells me later, loves to act the sommelier. What would his message to Lars be then? "Come," Tom says. And we all dream about how Lars could do it: Mission to Tom.

Mission to Lars tours Picturehouse cinemas through June and July