I'm not sure Britain actually became a diverse society: I think it's always been that way.
Consider their history. The ancient Brits were a diverse lot, comprising many tribes and groups. Then the invasions started and they never had a chance to consolidate. So the British stock was fleshed out by the Vikings, Normans, Saxons and so on. And, of course, there was a pesky group from Rome who left behind a big footprint after they went home.
The English weren't good neighbours and incurred the ire of the Scots, Welsh, and Irish before those people eventually compromised (that is, they were subdued) enough to form the United Kingdom. That is still not a comfortable fit for many of them.
Rural England (more so than Scotland, Ireland and Wales) developed insular communities that not only spoke English with different accents but added local words that would have sounded obscure in other parts of the country.
So there was cultural (and linguistic) diversity which broadened with the growth of the British Empire. In time people from the empire were allowed to settle in the UK and that changed things still more, as did modern immigration.
Further change came with the EU. Britain's borders, with the help of air transport and the Channel Tunnel, became even less clearly defined.
"It is time for parents to teach young people early on that in diversity there is beauty and there is strength." -- Maya Angelou