You seem to be confusing two different issues, which is falling housing prices in some areas, whilst rents continue to be high.
In London and the South East housing prices have been inflated largely by overseas investors and the vicious cycle created with them investing in luxury housing developments. Not by immigrants because these immigrants absolutely cannot afford to buy housing in these regions - they rent.
That boom really picked up after the financial crisis, and as I said, was predicted to burst from any time from 2013 onwards. It slowed in 2015, and has finally reversed. This is kind of shit for everyone. Housing prices are still too high for actual people with actual jobs to buy in these areas, but are dropping enough that overseas investors are losing their investments, and they’re not going to ever drop to affordable levels. Companies developing luxury housing are going under, and no one is building sensible stuff, which is going to put builders out of work again and we’ll be back in a bad situation, with councils still prevented from building much by various Tory restrictions. As most MPs are multiple home owners they have personal conflicts with any real fix as it would negatively impact them. Teleworking is going to make the demand drop further ensuring the situation gradually gets worse, whilst there’s just enough demand to keep prices high. There are also big problems with service charges, which, hilariously, are often both locked in by contract and too LOW for the way luxury flats have been built. The companies and associations which own they will run out of cash and those luxury flats will then decay. They’ll need to be knocked down because the builds are simply too fancy to maintain (too much glass, fancy use of wood, etc.).
Immigrants, as I said, rent, but typically share properties and pack people in. As a result they could pay fairly high rents. As they move out, rents stay high because the landlords are locked into mortgages they took out specifically to buy properties intending to rent them out. A lot of them are also trying to sell before the market goes low enough that their mortgages are a big problem, further limiting real rental stocks.
The only real solution is the same as it was in 2010. Building affordable housing. Councils are prevented from doing so, the government refuses to do so for ideological reasons (whilst constantly acknowledging the problem!), and private businesses can make enough money so won’t.