Sadly, this problem isn't unique to London, but I'm not sure I would characterize it as a "rich people problem." It's a regulatory problem. I assume in London it's the same as NYC, LA, Miami, etc., where foreigners are buying up real estate as money shelters for their stolen gains in their homeland. Saying, "the rich are unwelcome here" is non-sense, because the native rich aren't the ones causing the issue. The people who are driving the market haywire don't even live in the condos they're buying, so they probably don't care if they're welcome or not.
What these cities need is strong ordinances that discourage using housing stock as a money shelter. I suppose punitive taxes is the most efficient way of doing that. And it would have to be very punitive, because if you can drop $10,000,000 on a place you might visit once or twice, then you aren't going to mind a few percent excise tax. Once the investment doesn't look like an investment anymore, then even a really rich person isn't going to waste their money on it.
I'm curious what the landscapes of these cities are going to look like in 20-30 years. This looks an awful lot like an unsustainable trajectory, so the price will probably collapse sooner or later. Then what? I doubt they'll be redistributed to the needy. Municipalities should wake up to the fact that sacrificing the future of their city so that some developers can make outrageous short term gains is not a good strategy, even if it fills their coffers now. My fair city, Detroit, should be an object lesson in what happens when you do no long term planning for the sake of satisfying a specific industry.