“Oh my ears and whiskers, how late it’s getting.” The White Rabbit’s words must have turned to poison in the ears of the collaborators behind Alice’s Adventures Underground, an ambitious immersive experience.

On press night, a power cut delayed everything by about an hour. A stopped watch is right twice a day, but the time stamp on my arm never fully recovered.

A swift start is essential for any production but none more so than in the Vaults. For Alice, a collaboration between Les Enfants Terribles and Emma Brunjes Productions, is a piece of precision engineering, requiring 56 visitors (divided randomly into playing card suits) to find separate journeys through a labyrinth of decadent rooms, seeing fragments of a story before getting shuffled back into a pack for the finale. Less theatre, more air traffic control.

“I’ve lost my husband,” said one. “He’s a Diamond, I’m a Spade. We got separated.” Which sounded much like a febrile actor but was, in fact, a visitor like me. (Although the line is often blurred – this is immersive theatre where you’re whipped into minor roles whether you like it or not.)

It is perhaps testament to this evocative wonderland – amazing costumes, fascinating interiors, other-worldly air – that initial misgivings are overcome.

There isn’t a narrative as such but each tableau – a bit of circus, or puppetry, or acting – adds something new, like a morsel in a nonsense buffet. The book is, after all, a series of non-linear sketches, says writer-director Oliver Lansley

The initial desultory shambling was less a flight of fancy than a horrible flashback to a recent airport experience. “If Disney did passport checks,” I thought as I trooped from holding pen to queue, ready to be processed again.

I found myself at the back so the whole Eat Me-Drink Me choice of storyline was lost. I was handed a bottle and shepherded urgently down the Drink Me corridor.

My particular patchwork involved Tweedledee and Tweedledum as stroppy trapeze artists, a spaced out caterpillar, shenanighans in the kitchen, a wily Cheshire Cat and a jittery knave.

This culminates in a monster tea party in the most impressive and outlandish of the rooms followed by a trial conducted by the Queen of Hearts where the threads of the story are sewn together in quite an effective and affecting manner. My Spade was happily reunited with her Diamond geezer too.

Afterwards we were released into an exotic themed bar and, beyond, the South Bank, which was inevitably mundane in comparison. The spell has worked and now is broken.

The creativity behind this project is evident, the scope audacious, and the marriage of imagination to spreadsheet bizarrely compelling, like a game of Tetris played with a Christmas biscuit assortment.

A quick squirt of WD40 and I can see this becoming a regular resident in the Vaults.

The Vaults, until Aug 30, various times, £35-£47.50