Vacant properties in London are a red herring. Last time I checked official number they accounted for 1% of the housing stock, bearing in mind that a number of properties are always being refurbished at any given time. In general it does not make sense to keep a property empty unless perhaps it is an extreme luxury one.

South Kensington is one of the most expensive area of London and not representative at all. Whatever happens there does not affect the average person, and quite of few of the grand houses there are also actually offices.

In any case, it is not true that 30% of homes are empty there. Apparently the City of London (i.e. the square mile, not London as a whole) has the highest rate in the country and that is still only about 4% [1].

In Cornwall the issue is quite different: The area is very attractive for second homes and, at the same time, it is the second poorest region in all of Northern Europe. This means that local resident have a very low purchasing power and indeed cannot compete with people from London and the South East. I think that they are starting to implement restrictions in some areas in order to give priorities to buyers who will occupy the property as their main residence in order to keep second homes in check.

[1] https://www.cityam.com/15bn-worth-of-london-homes-sit-empty-...