It’s a blustery Saturday night and around 900 people are lined up to go to church. The faithful have come to the Union Chapel, in Islington, London, to receive wisdom from those stepping up to the pulpit – a lineup that tonight includes Benedict Cumberbatch, Noel Fielding, Juliet Stevenson and Thom Yorke. Letters Live, a celebration of literary correspondence, has always looked a little odd on paper since it began in 2013. After all, who wants to spend a night out having letters read to them by a bunch of actors and musicians, whose identities are not revealed in advance?

“When you first hear the idea it doesn’t quite sing, but do it once and you are smitten,” says Cumberbatch, who was so smitten he became a producer and now helps shape the shows. “Everyone we ask says they’d do it again in a heartbeat. And we just hope it is the same for the audience.”

Jamie Byng adds gleefully: “When I told Toby Jones about the concept, he said, ‘What’s so fucking exciting about reading letters?’ Now he loves it! He’s proselytising!” It helps that Byng, as the head of Canongate, can just call up Toby Jones – or, as he did for the first ever event, Nick Cave. Now the show that was once a book – Shaun Usher’s Letters of Note, which in turn grew from his hugely popular website – recently sold out three gigs in less than an hour, a feat totalling 2,500 tickets. And today, it has announced its biggest ever venue: the Royal Albert Hall.

Part of the allure of Letters Live is the nightly celebrity bingo: Gillian Anderson reading Katharine Hepburn, Ian McKellen reciting Roald Dahl, Jude Law at a show in the Calais refugee camp. New York events have seen James Earl Jones, Brie Larson and Chevy Chase on stage. There are great moments of verisimilitude – Cave reading his own letter to MTV, or Ben Kingsley reading the letter Gandhi wrote to Hitler, 32 years after portraying the Indian activist. Last Friday, days after her Oscar win, Olivia Colman materialised to a standing ovation, to read a letter by Queen Victoria to her eldest daughter. “Pressure’s on,” she opened, drily.

“You can’t help be drawn in, even when the letter is never written for you,” says Byng. “Every human experience can be found in letters. They can make you feel something you can’t feel yourself, experiences you will never live. Some of us will never lose a child. Some of us will never kill someone. You could sit at home to read them, but, through the alchemy of performance, everyone in that audience becomes the recipient.”

Aside from the star factor, a huge part of why people go to Letters Live is an ineffable longing for a return to more nuanced, intelligent communication. There are always momentous letters – Buzz Aldrin reflecting on his experience on the moon, Virginia Woolf’s final letter to her husband before her suicide – and there are very ordinary lives on brilliant display, like the irate Australian woman who wrote to her neighbour about his playing Daft Punk at 5am, read with great gusto by Noel Fielding: “As I was awakened by Pharrell Williams’ dulcet tones, Nile Rodgers’ signature rhythm guitar and the on-point (although highly derivative) production of Guy-Manuel de Homem-Christo and Thomas Bangalter, I was torn … I thought to myself, ‘Maybe these inconsiderate assholes could display the modicum of consideration for their neighbours, recognise that most of the building is asleep at this time, lay off the pingers and shut their fucking windows.’”

Fielding is particularly taken with the Australian slang for ecstasy. “Ben, you got any pingers?” he hollers after the show, to which Cumberbatch winces like a despairing father. The whole production has, in the very best sense, all the last-minute feel of a school play: new letters are handed over backstage, and the stage curtains bulge as other readers gather in the wings to listen in. After all, the letters are, says Cumberbatch, “some of the best material an actor will ever see”.

“I was crying laughing listening to Noel read that Daft Punk letter because it wasn’t just ‘Shut the fuck up’ – it was the absolute alternative, which is so beautiful,” says Thom Yorke, fresh off stage from reading a letter to the Guardian about the French burqa ban. He adopts a lofty tone – “‘The on-point, although highly derivative production’ – oof!” He mimes taking a blow.

Yorke “didn’t really understand” Letters Live when Cumberbatch first asked him along, but he’s a fan now. “I feel there is a recuperative effect in listening to people’s letters. In this day and age it is quite easy to be the cynic, but it reminds me of the olden days, of writing faxes to my friends. Letters are inherently the most considered, self-aware way for humans to express themselves – you have to measure your emotions in a way you don’t when you use Twitter.”

Fielding says: “When I am on, this is Fisher Price Letters Live. If you get a serious letter, you can’t muck about, you’ve got to nail it. They don’t give me letters about famine or the environment for a reason.” He marvels at the Shakespearean types taking on grand letters. “It is so beautiful, there is no way I could do it like them. I am always surprised they ask me back, but it is one of my favourite things to do. There’s nothing else like it.”

As a producer, Cumberbatch is wary of giving the audience some fun (Fielding, pingers), while also putting more spotlight on letters that reflect our current state of affairs, like Kurt Vonnegut’s Ladies and Gentlemen of AD 2088, a frighteningly prescient warning of climate change penned in 1988. The mood can shift page to page, night to night: one woman in the audience tells me she’s been two nights in a row; another was so moved by Colin Firth’s reading of a grieving widower’s letter to the Bataclan terrorists, years ago, that she wells up at just the memory.

It is this intimacy that Cumberbatch is most worried about preserving in a space as big as Royal Albert Hall. When asked about the show, he involuntarily touches wood, gripping a nearby cabinet, then frowns at his hand. “I don’t know why I did that. I think we’ll be OK,” he says.

The Union Chapel show ends with another standing ovation, this time for a letter sent just two weeks ago by teenage climate-change activists, demanding action from the world’s politicians. Two of the authors, Anna Taylor and Ivi Hohmann, read it with passion, before promptly being enveloped by their fellow cast.

“We need to use these events to address the world right now. Short-termism is the greatest flaw of our species. All these letters are little grenades we can throw into the audience. Because all of it is reasoned argument, which is not always found in debate today.” He smiles. “Except between angry Australian neighbours.”

• Priority tickets for the Letters Live show at Royal Albert Hall on 3 October are on sale now. General sale opens 15 March. A new series of themed Letters of Note books by Shaun Usher will be published 3 October (Canongate).