Seeing how short all the buildings are in London, its housing shortage is obviously fairly self imposed in that city.
Lack of housing is almost always a zoning issue. Builders will always build if there's money to be made and people to sell houses to. The only reason they're not is because they're not allowed to.
There are a ton of skyscrapers being built. They’re just not particularly affordable
There are plenty of tall new buildings being built.
So many commenters about London on here who don't actually live here.