Since supply of house cannot be increased to solve this problem, you need to lessen the demand. The most obvious way to do this that I can see is to put a cap on immigration that is much lower than the number of people leaving (about 400,000 people leave the UK each year). However for various reasons this is seen as absolute verboten.
BTW, I know exactly the type of buildings you are talking about (we have them in Manchester) and they are typically look awful and usually start falling apart after shortly after construction. They are also not very nice to live in (I have lived in one for short amount of time).