Illegal immigrants aren't directly relevant to the housing question under discussion here.
As for the numbers, there are lots of ways to gather data and much research has been done. The range of estimates is more down to how you define illegal i.e. do you include those in the asylum processing system, student and holiday visa overstayers. Based on generally accepted definitions there are probably about 600,000 people in London who are undocumented and maybe around 300,000 in the rest of the country.
While these numbers are too large, these people have no recourse to public funds, are largely gainfully employed increasingly in the gig economy and tend not to be involved in criminal activity other than that related to their immigration status. American is finding out that trying to clear undocumented workers out of the country isn't a pain free or economically beneficial exercise and I suspect that this is why there has been no concerted action in this country on the issue. |