London is very very international. People are here because they are well educated and have well paid jobs. They want their own children to go to internationally recognised universities either here or overseas.
Whatever the domestic reputation of the University of Salford or wherever, it is really just Ivy League, London or Oxbridge that cut it, with the odd European, Australian or Canadian University thrown in.
Once the grammars are excluded most state schools in London don't deliver at this level.
People may also be here because, post-Arab Spring etc, London has become not just a place to have a pied de terre and to visit but to have your family home and to educate your children. Think Abramovich.
In DDs main out of school activity in Central London she has worked out that only 10% of her peers have two British parents and virtually all are privately educated. Similarly both DC have friends whose mothers have few links to the UK but whose fathers travel massively. If you get your children into a good London school you might as well be based here as in Dubai, Hong Kong or wherever. Fluent English is the bonus. IB is then popular for those heading abroad for University.
It is no coincidence that house price rises in Central London show no sign of abating. People are still coming in. Their children still need to be educated.
I am not sure the extent to which growing up in such a cosmopolitan area affects the children's sense of being British. I assume like many, they see themselves as Londoners first and British second. I like the way that my son's school really promotes school trips within the UK. I think the assumption is that pupils will have been all over the place, but probably not to Wales or the Yorkshire Dales.