These are neither purely traditional nor fully globally assimilationist.
Singapore, New York, Los Angeles, San Francisco, Seattle, Austin and San Antonio, TX, Vancouver, Sydney, London, Berlin, Miami, and Rio all come to mind. Those are among the larger cities, there are also smaller and more outlying regions.
People have always moved, migrated, and interacted. Quite often peaceably, as the alternative is extraordinarily destructive.
You need to brush up on your history my friend. The number of times the island has changed hands runs counter to your point, and the current inhabitants certainly aren’t some continuation of an ancient people.
Rome held it, the Britons held it, the Picts invaded from Ireland, the Angles and Saxons had it after beating the Picts and killing the Britons…the list goes on. There isn’t some sacred ancient culture at stake and I’m sure the groups that were kicked out by the current inhabitants would take umbrage with your point if they could speak up.
I’m not even going to dive into “London is a miserable failure.” That’s just another hand wavy talking point. I’m not even sure by what metrics you’d measure how London is a “miserable failure.”
Is the problem that immigration drags down economies?
Or that it leads to gentrification?
You know, all those lazy Polish plumbers coming over and stealing all those British jobs....
Sounds like you've found a cause for every problem you see.