I think you're thinking of immigrants as the people coming to the UK from poor countries. Most of those people don't go to London, they move to regional towns and cities.

The people who move into central London from overseas are rich emigrants (Russian oligarchs, Saudi princes, etc).

There is significant social housing in London, where the local council provides housing for people of limited means. This is a historical leftover from more enlightened times (and one that Thatcher tried to eliminate). But the amount of churn in this housing, as you can imagine, is very low - once you have a council house, you never leave it because all the other options are waaaay out of your price range. So there are whole families who have been living in poverty for generations in the centre of London.