"Those at the bottom see their wages driven down, rents increased and have to compete for any housing at all and for work."

The wages are driven down by British employers applying "market forces". The rents are driven up by greedy British landlords who bought their houses with favourable terms determined by the British government and who are entitled to a lot of bogus "allowances against tax". Housing shortages are the upshot of British government policy over decades. The ones at the bottom are competing for part-time jobs, jobs with no security, zero-hour contract jobs and bogus apprenticeships, all British government policy eagerly snapped up by British employers. Of course, you can always declare yourself bogus self-employed, which is another way of saying employment that doesn't require British employers to pay national insurance, again encouraged by the British government. Oh gosh, sorry, I don't appear to be able to blame immigrants for anything...