London's basically a broken city. Over the last thirty or so years, it's essentially become a place where you can be very well off, or relatively poor, and it's very difficult to be anything in between.
Immigrants aren't driving out people who were born and raised in London. House prices were pushed up so high that many Londoners will have had to move out. At the top, there are the lots of people in highly paid jobs, and plenty of foreign born people who are based in London, as it's one of the top few international cities in the World.
The housing for average earners - where we would be in a semi-detached, or a terraced house around Manchester just doesn't exist. I worked for charities when I lived there, and as recently as 1999, when I bought my first flat, the price was reasonable compared with my salary. By the time I was selling my last place to move back to Manchester, in what was one of the few "cheaper" areas, the people viewing were earning 3 to 4 times my salary.
So instead of average earners, you get huge numbers of people house sharing in poor quality accommodation - something that attracts two main groups - young foreigners from Europe, Australia etc., who are happy as they just want to be "in London" - and immigrants from poorer countries, who are doing all the service jobs that the well off need.
And that social housing you hoped to get? In London it was messed up in the 80s with right to buy. Because it was becoming so valuable, much of the good quality housing stock was bought out, often by speculators funding the purchase for people who couldn't afford it on their own. What was left, tended to be the poorest quality houses in the roughest areas, and the tower blocks, which had huge maintenance costs (and of course LAs weren't allowed to use their right to buy money to keep these properties in good condition). This leads in two directions. First, that the housing stock is really low quality, and not in great areas, so again, it attracts people with little choice about where to live. Second, as houses are sold off, the pressure of numbers means that you have to be either statutory homeless, or disabled, to have any chance of rehousing. So, you end up with areas becoming even more impoverished.
London has definitely changed, and that's reflected in the number of immigrants, but the driver of the change came from the wealth pushing people out, rather than the immigrants.