None of which happens.
Take a look at what migrants need by way of qualifications, experience and salary to get a visa. I wouldn't call the salary requirements "cheap labour". I wouldn't call them low or unskilled labour either.
Immigrants rent already empty properties. No one us being made homeless.
When you say "looking after old people", I guess you mean working in health and social care. You are so wide of the mark if you believe looking after old people is all that is all about.
However, a large percentage of migrants do work in health and social care. Without whom, many hospitals, care homes, supported living locations, domiciliary care locations and residential care locations would be woefully understaffed.