Fans of Back to the Future – and I sometimes think that as a group we have as large a diaspora as a major religion – take the business of liking this time-travel movie very seriously. We have to. Fandom is usually early-fixed, incurable, big on conversationally disruptive quoting. Whether aged 14 or 40, a particular dashboard speed can never be reached without an instinctive ululation, "88 miiiles per houuuur"…

When Secret Cinema, an organisation that arranges ambitious, mixed-media screenings of films, announced it was to show Back to the Future over 30 nights this summer, it sold more than 65,000 tickets. (On the day the project was announced I was on holiday, and received such a bombardment of texted BTTF quotes from fellow obsessives I assumed a cast member had died, if not that someone really had invented time travel.) The last Glastonbury festival sold 120,000. Secret Cinema had scored a big achievement, especially as its screening-with-extras cost £53 a head. It's those "extras" that give Secret Cinema its zing: live performance, music, food, dress-up – all sorts of tricks layered over a main-event screening, the details of which are kept a secret until the last moment.

When I was invited, last week, to tour Secret Cinema's Back to the Future screening site in east London, a hush-hush visit ahead of Thursday's opening night, I was excited. "Great Scott!" was the sort of thing I repeated to myself on the way there, and "The flux capacitor is what makes time travel possible!" I walked around a plausible 1950s world created by Secret Cinema (this era chosen because the film's main character, Marty McFly, travels back from the 1980s to 1955) and saw imported old cars, boys with oiled hair, girls with knotted shirts, stunts being rehearsed, trees being sheared.

I also signed a long non-disclosure contract that would stop me revealing more than these vague details, the organisers insisting, fairly, that they wanted future audience members to visit the site with the capacity to be surprised. Then I sat tight, along with about 3,500 opening-night ticket holders, to wait for Thursday.

At which point, a hiccup. You might have read about it in the Guardian, on the BBC, on Twitter, on BuzzFeed. The show was cancelled, two hours before it was due to start, leaving hundreds of fans dressed up in 50s gear with nowhere to be called Daddy-O.

On Facebook, there were itemised complaints about time taken off work, hotels booked… People said they'd flown in from Israel, the US, even Cambodia. Secret Cinema said that the cancellation was due to "extenuating circumstances".

"Back to the drawing board," an upset customer wrote. Another, tweaking an obscure bit of the film's dialogue, said: "I'm afraid you're just too darned disorganised." (The sign of a true BTTF fan, I thought. Complaining by quote.)

There was a lot of anger sloshing about – then, while writing this, Friday's, Saturday's and Sunday's screenings were also scrapped. Who knows, now, if it'll ever happen… Could one of you time-machine it back from the future and let me know before my deadline?

Secret Cinema is run by a 39-year-old maker of short films, Fabien Riggall. He first began staging screenings of shorts in London in the mid-noughties, interspersing footage with bands and DJs. The idea was to make the viewing of shorts more dynamic. Putting on full-length features was an obvious next step. About 400 people came to his first Secret Cinema show, in 2007 – Paranoid Park projected in a moody space under London Bridge. Since then, more than 40 films have been given themed treatment, foam pies thrown at Bugsy Malone, Rydell High built for Grease.

Ticket sales have risen steeply (15,000 for Lawrence of Arabia in 2010, 30,000 for Prometheus in 2012) as has their cost. Demands on the audience have been on an incline, too, and it must have helped to have a play-along attitude when front of house staff asked new arrivals for their clothes before watching The Shawshank Redemption. (People were given prison jumpsuits instead.)

"We work hard to make the audience part of the performance," Riggall told me as he walked me around the Back to the Future set. He had wanted to screen this film for years. "Back to the Future represents the idea of Secret Cinema in so many ways. It catapulted people into another world."

I found the attention to detail at the site – a large outdoor compound – extraordinary. Many elements of the set had been spun out of background glimpses from the film, references you'd only register after an unhealthy number of viewings. Riggall was clearly a fan, an auto-quoter; also a perfectionist, bothered, as we walked, that the writing on the side of a yellow school bus didn't look quite 1955 enough. He pointed out an old-fashioned garage that was going to be staffed by a real mechanic, and a working analogue telephone exchange. "It's not just a set," he said, "I want it to work." Though Riggall didn't want to say a total cost, a small fortune had obviously been spent importing tents and picket fences, hiring animals.

I was impressed by Riggall's ambition, by the depth of immersion he seeks. While the multiplexes seem to be racing to make filmgoing expensive and unglamorous, here was romance. "People come to Secret Cinema in their thousands because it takes them out of this robotic existence we have," Riggall said. "I think the world should be unpredictable, mysterious, romantic." Secret Cinema's aim was to offer that, one screening at a time.

But romance is risky. Choosing a beloved film such as Back to the Future opened up Secret Cinema to unprecedented sales and a massive new audience, not all of whom, I expect, would have willingly surrendered their clothes for Shawshank, and not all of whom were forgiving when week one of Back to the Future was cancelled abruptly.

Who knows? The opening-week fiasco might wind up as a watershed, the point at which a mass audience stop risking larger, pricier accompaniments to their evening film than ziplocked Maltesers or glasses of Picturehouse red. I hope not.

"People want experiences that are not just 'the traditional experience'," Riggall told me emphatically. It was one of several things he said that, had the on-set time machines been in working order, we might have known would take on extra meaning later in the week. But I think he was still right: that the traditional cultural modes are worth prodding, stretching, upsetting.

In one of Back to the Future's climactic scenes, Marty McFly takes to the stage at a high-school dance, there to impress a room of 1950s teenagers with a medley of music from the future. After some rock'n'roll – they love it – he segues into 80s metal. Hands clamp over ears. Marty has rushed ahead. "I guess you guys aren't ready for that yet," went the line in the film, "but your kids are gonna love it."

Secret Cinema presents Back to the Future until 31 August; details: secretcinema.org/tickets