Alice's Adventures in Wonderland, Royal Opera House, London

The Royal Ballet's new Alice's Adventure in Wonderland is big and glossy, sharp in execution as the Queen of Hearts' axe. For much of its length, it's a show more than a ballet. Christopher Wheeldon's fluent choreography sometimes has to fight for attention with the wit of designs and staging, helped by Joby Talbot's exuberant score.

There's a lot riding on this Alice, the Royal Ballet's first new evening-length work this century. Wheeldon, one of ballet's most in-demand choreographers, has said he wants to give this generation of dancers their own story ballet. He's chosen a tricky subject: this story isn't obviously danceable. The self-possessed child heroine falls down a rabbit hole, meets curious creatures and wakes up. Much of the point is in the language: puns, riddles, invented words.

Wheeldon and playwright Nicholas Wright, who created the scenario, don't pretend that Wonderland is anything but episodic. They add a framing device, with Lewis Carroll visiting Alice's family in Oxford. Characters from Alice's waking life – including her friend Jack, the gardener's boy – will reappear in her dreams.

Any production of Alice has to deliver Carroll's surreal transformations. Working with Talbot, designer Bob Crowley and projection designers Jon Driscoll and Gemma Carrington, Wheeldon makes a brilliant job of the famous changes. Edward Watson's Lewis Carroll sprouts the White Rabbit's tail, then jumps through his camera bag into the rabbit hole.

A puppet Alice dangles above the stage, falling and landing as Lauren Cuthbertson's real girl. Music and projections swirl and shimmer, doors grow and shrink. Cuthbertson's tantrum solo is gleefully danced, full of frustrated jumps and angry little kicks. Alice chases a tiny door across the stage, trying to squeeze through it to reach the garden – which blooms out in the auditorium.

Wheeldon and Wright keep too many of the book's incidents: the caucus race flags. There's more energy in the Duchess's terrifying cottage, which Crowley designs as a "Home Sweet Home" sampler outside, a gory butcher's shop inside.

Actor Simon Russell Beale dances the Duchess. It should be glorious casting, but even he has to push for attention against the whirl of the staging. (He makes more of an impact as a party guest in the first scene, alarming in a bonnet.) Steven McRae's Mad Hatter tap dances valiantly, but his personality is muted by so much costuming.

Wheeldon has fun with Busby Berkeleyish corps dances, and makes elegant duets for Alice and Jack, now transformed into the Knave of Hearts. Sergei Polunin's Jack soars effortlessly into his steps. Talbot lets rip with music for a playing card ballet; the choreography is too polite to keep up. In dance terms, the show is stolen by the Queen of Hearts, the monstrous dream version of Alice's mother. Zenaida Yanowsky stalks through a parody of The Sleeping Beauty's Rose Adage, partnered by cavaliers who are terrified they're going to be beheaded. She shimmies triumphantly, seductive and hilarious

Other characters are less well-defined. Alice leads us through Wonderland; she doesn't dominate the work. That means Wheeldon's Alice is less likely to be its generation's ballet – but it's a dazzling show.

Until 15 March