Source on the "plenty" of empty flats in London? The number I found is 29,242 which is hardly something that'll get the prices down.

Indeed, and vacancy numbers are to be expected. Go to any source for rental investments and you'll typically see vacancy rate assumptions of 4% to 8%, because tenants leave and placing a new tenant can take some time. If you plot that on a city home to 10 million people, it's entirely natural to find that a few hundred thousand homes are vacant at any point in time, typically only a few weeks or months. The 'long-term vacant' problem is relatively minor by comparison, and while that must certainly be addressed, it's not at all going to change the order of magnitude of the current housing issue.

London's vacancy rate is actually very low, even during covid when vacancy rates tended to rise.