The empty houses in London make up a tiny percentage of the entire total, and are usually foreign owned, bought with cash.
Whilst its true that schemes like help to buy have driven prices up in places like London, the biggest thing driving house prices is under supply.
For nearly 40 years the private sector have been under building by about 150-200,000 needed dwellings a year. This drives up the price of new builds which suits them down to the ground.
When you look at the failure of the housing market there are several different causes, but the under building is the major one. For example the ONS calculated that out of the last 23 years, immigration had increased house prices by 20%, but during this period house prices had risen 320%. So immigration added 0.86% PA to the price of an average house but 13.04% was from somewhere else, the significant part of this is under supply.