I remember the exact moment I realised that Mischief Theatre had become a big deal. Travelling across the Paris Metro last year, I caught sight of a poster for a play called Les Faux British.

On it, a tartan armchair was ablaze, one spring popping out of its upholstery. A portrait of a corgi was hanging at a jaunty angle behind it and the ‘h' had fallen off the title.

What the hell was this? ‘Une vraie comédie catastrophe.' Then I saw the creators' names: Henry Lewis, Jonathan Sayer et Henry Shields. Aha. Une Vrai Comédie Catastrophe: The Play That Goes Wrong.

On New Year's Eve Mischief Theatre made their television debut (or ‘debue', to use the Cornley Polytechnic Drama Society's own pronunciation). Their second show, Peter Pan Goes Wrong, was remade for the small screen, shot at Pinewood Studios with David Suchet joining the fun.

It's a fine way to kickstart what will be a big year for Mischief Theatre. In March, they'll open The Play That Goes Wrong on Broadway produced by none other than Star Trek director J. J. Abrams. It's on course to go global: an Australian tour starts in February, and there are supposedly more than 20 other productions lined up worldwide, with translations ready to go from China to Brazil.

It's the next chapter in an extraordinary Cinderella story. Lewis, Sayer and Shields are the youngest writers to have three shows playing consecutively in the West End. The Play That Goes Wrong is still running, two years on, at the Duchess Theatre, having played to a quarter of a million people. Peter Pan Goes Wrong is doing a seasonal stint in the Apollo, and their latest, The Comedy About a Bank Robbery made several end of year lists this month. Don't be at all surprised to see Mischief Theatre pick up their second Olivier Award next year either. (And don't forget that you lot, WhatsOnStage readers, gave them their first gong back in 2014.)

Not bad for a show that opened to an audience of four. Yes, four.

The Play That Goes Wrong
The Play That Goes Wrong
© Alastair Muir

Rewind four years and The Play That Goes Wrong wasn't even called that. It opened as The Murder Before Christmas at the Old Red Lion, a tiny pub theatre in Angel, while its writing team were still working minimum wage jobs: waiting tables, pulling pints, cold-calling strangers.

I saw its second incarnation, downstairs at the Trafalgar Studios in 2013. By then, they had the title, but not the finished show. It was, at that stage, still fairly tatty. Cramped onto a small stage, surrounded by shaky wooden flats, it lasted just over an hour and seemed to me to be a knock-off Noises Off: the same joke, just without Michael Frayn's layers. It was, I wrote, "like a first draft Frayn might have scrawled on a napkin."

To a certain extent, I stand by that – but then Frayn's farce is a bona fide masterpiece, one of the all time great plays. What Mischief have ended up with, however, is a fine piece of straightforward slapstick – and it's largely thanks to producer Kenny Wax, who spotted the potential that passed this critic by.

West End audiences aren't seeing the same Play That Goes Wrong, you see. It's bigger, longer and, er, wronger these days. Far, far funnier too. With a proper set and an extra 45 minutes, it spirals into something wild and unhinged, something unpredictable and unstoppable, even, if it's not too much, something profound. By the time a grandfather clock is standing in for a corpse laid out on a chaise longue, nothing is what it purports to be. That's theatre, but equally, that's life. Ludicrous. Off the rails. Nonsensical.

You get two questions on repeat as a theatre critic: ‘What should I see at the moment?' and ‘Have you been to The Play That Goes Wrong?' More than anything else in the West End, it's the one show people want to recommend. Of course it is. As tips go, it's a cast-iron cert: a show that transcends age, class, even language; a proper old-fashioned family show. Neither too clever for its own good, nor too dumbass, it simply takes aim at the funny bone and bashes it about for a few hours.

While Mischief Theatre may not hit the sort of giddy heights that leave you helpless with laughter, they've created something fundamentally crowd-pleasing and egalitarian. If it's a throwback, well, that only seems to chime with the mood of the moment – our newfound taste for nostalgia and national pride. Mischief's slapstick speaks to something deep in our national DNA. Its ancestors are Morecambe and Wise, Norman Wisdom and Charlie Chaplin.

That's part of the reason I'll be fascinated to see how they fare across the Atlantic, but the other is they arrive there not as the plucky underdog – that surprise snowball of a success – but as a garlanded transatlantic transfer. That doesn't always go so well. Just ask Hand of God, the off-off-Broadway smash that bombed in the West End, but, where Robert Askins' sock puppet play was all cynicism and snark, The Play That Goes Wrong is straight-up, sweet-natured silliness. And it has Mischief in its favour.

The Play That Goes Wrong continues to run at the Duchess Theatre, Peter Pan Goes Wrong runs at the Apollo until 29 January, and The Comedy About a Bank Robbery is currently booking at the Criterion until 23 April 2017.

Peter Pan Goes Wrong is broadcast on BBC One at 6.20pm on 31 December 2016.