Granted, yes, we had little control over migration from the EU from 2004 to 2016. But very few of London's working poor are of Eastern European heritage - anyone from there who finds it too difficult here tends to head back.
Asylum seekers / "small boats" migrant numbers are absolutely tiny relative to the London population.
London does have a high % of first-generation immigrants - but most of them are now-elderly Caribbeans, South Asians, Mediterraneans, West Africans who came over in the 60s and 70s. Outside of a few areas, non-Anglo-looking people in their 30s and under were overwhelmingly born here.