The core question here, which I genuinely don't know the answer to, is how much of the supply crunch is being caused by the additional properties of individual parties? Are there really that many second, third, etc. homes in the UK?
Just from some quick looking, it seems like London's residential vacancy rate is something to the effect of 3%, so I'm inclined to think that the core problem is more lack of total supply rather than poor allocation of existing supply. We've seen cities boom all over the world in the past 20 years, but new construction hasn't remotely kept up with the pace of population growth. This is the pretty inevitable result.
Edit: Just to add numbers, London's population has grown by nearly 2 million since 2000. I would be highly skeptical that two million new apartments have been added in the same time. Add in the fact that a lot of those 2 million people are educated high-earning professionals, and this is what necessarily happens. When demand exceeds supply, price goes up. When the people driving that demand are relatively wealthy, price goes up a lot.