London may be a hot and flustered Olympic host this summer, but a small, perfectly formed exhibition celebrates our inarguable global significance in one important cultural respect: Shakespeare. Shakespeare: staging the world [2]forms a wooden "O" in the centre of the British museum (design by Tom Piper and Alan Farlie of RFK Architects). Curators Jonathan Bate and Dora Thornton have panned for gold here, and the exhibition shines with their booty. As we prepare for the world to come to London, we find that 400 years ago the world was already here.

This inspired collaboration between the British Museum and the RSC exhibits some 200 objects, from the prosaic to the decorative, that link like enjambment to Shakespeare’s transforming imagination. We’re taken from a copy of the Arundel First Folio (1623) through a series of rooms relating Shakespeare’s plays to the world - or worlds - around him. It’s a deeply considered mix of connections and conjectures.

We see, initially, a little of Shakespeare’s contemporary reality, at a time when globes and globalisation were brand new. An early map misinforms as to the location of his theatre, muddling it with the bearbaiting arena. Easily done, at the time. Philip Henslowe, a sort of 17th-century Cameron Mackintosh, owned a portfolio of entertainments on Bankside, and the Globe sat alongside his bearpits and brothels; a she-bear’s battered skull on display mutely testifies to her brutal treatment in a violent world.

A painting of Richard III (“every tale condemns me for a villain”), suitably crook-backed and with broken sword, reminds us of the prevailing Tudor orthodoxy. Writing in a heavily censored milieu, for a ruling class jittery about conspiracy and succession, Shakespeare turned adroitly to the historical and classical past to navigate politically sensitive themes. On display, for example, is the gold aureus, coined in 43 BC by the Roman conspirators to commemorate the assassination of Julius Caesar (“the unkindest cut of all”). EID-MAR is clearly inscribed on the back.

Some objects really snag the gaze, like Henry V’s funerary “achievements”. These are the totemic armoury cum relics of the warrior prince, championed by Shakespeare as the apogee of patriotism (“we happy few, we band of brothers”). Shakespeare mentions the helm and sword in his play and undoubtedly would have seen them, exhibited as they were in Westminster Abbey well into the twentieth century. And here in 2012, you too can still see the delicate fleurs de lys woven into the shield’s Chinese silk, of palest blue.

Ironically, we have censorship to thank for the only surviving example of Shakespeare’s handwriting: close, inky, elegant.  Sir Thomas More, which he co-authored, depicted the 1517 London riots against immigrants (“strangers”). Unsurprisingly the Tudor machinery quashed any performances that explored the character of More, so the play was never printed and his handwriting survives.

We visit deepest, loamy England - some simple Elizabethan garden tools; a tapestry of Warwickshire and the Forest of Arden. This is the garlanded countryside of As You Like It and Shallow’s orchard in Henry IV. Stitched into the tapestry are the great houses of Midlands Catholics: the tensions between old and new ways are never far from his text’s surface.

Time and again Shakespeare ransacked other worlds to stage contemporary debate. We get some sense of the traffic of objects and people that was beginning to flow into London, and feed this imagination. The Moroccan ambassador, whose portrait hangs here, made quite a splash in London in 1600. We can only speculate how he fuelled the creative journey to Othello (“the noble Moor”).  A narwhal tusk from Frobisher’s voyage to Baffin Island, the painting of a Brazilian marmoset, form part of a “brave new world” of exploration that illuminates The Tempest.

We end as we begin, with a copy of Shakespeare’s works. This one’s a bit cheap and tatty. But it was the very book smuggled into Robben Island by prisoner Sonny Venkatrathnam. He adorned the exterior with Diwali cards and persuaded the authorities it was a Hindu bible. In it the ANC prisoners secretly earmarked and signed passages that resonated with them.

The book lies open at Julius Caesar. Here Nelson Mandela has highlighted and dated (16th December 1977) Caesar’s speech beginning “Cowards die many times before their deaths.”

This, say the curators, is an example of Shakespeare as global resource, and it’s difficult to disagree. It’s one Olympiad ticket you’ll not want to miss out on.

The BP exhibition Shakespeare: staging the world at the British Museum [3] until 25 November.