Multi-culturism seems to have acquired deep roots and flourished there long before now.
London has been a multi-cultural city for at least two thousand years - indeed, for longer than it has been in its current location*.
Once the Thames was established as a major trade route (prior to the Roman invasion, probably as early as the paleolithic), the multiculturalism of the inhabitants at locations where it could be forded or bridged was inevitable. The last fording point in the downstream direction was around the current location of the Houses of Parliament; The last location suitable for constructing a bridge was around the current site of London Bridge.
Inevitably the river just downstream of these points would become a major port; Equally inevitable was the the North bank would be chosen, as the south 'bank' was marshland with no clear boundary between river and swamp.
As the chief port of Britain, with boat access to both the interior of what was to become England, and to the wider world, London has been a multicultural settlement from the very beginning.
The Roman Empire brought people from what is now the Middle East and North Africa, and they never left. The Angles, Saxons, and Jutes all rolled up and added to the diversity. The Danes (who traded from Scandinavia to the Mediterranean) brought people from across Europe and from North Africa. The Normans injected a French seasoning to the mix, and the Crusaders brought further Middle Eastern people in to the city. Medieval trade through London was huge, and it became Europe's largest port, which it would remain until the C20th. The 'soft' protestantism of Henry VIII made England (and specifically London) a haven for refugees from Spanish Catholic totalitarianism, particularly Jews (though refugees had been arriving since well before Henry's reign); The Spanish colony in what is today The Netherlands was particularly convenient to London, and there was a huge flow of refugees across the Channel. England began her explorations of the world, inspired by the Spanish and Dutch success, bringing still more people to London from the Anericas, from India, and from the Far East. The Thirty Years War drove another wave of European refugees to London, and a lot more Dutchmen arrived with William of Orange at the close of the C17th, bringing with them people from the new Dutch colonies in the Far East and the Americas.
And then,
after all of that, the British set about building an Empire that spanned the globe, with London as its centre...
*Between the departure of the Romans and the arrival of the Normans, Roman London was effectively abandoned, and the population centre drifted to the area now called Westminster; It returned to its Roman location with the construction of the Tower of London by William the Bastard).