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.
You’re often not even allowed to build things that aren’t in keeping with the area even if nobody can see the bits of the property. It’s mostly about stopping people nearby having a nicer house than you.
You could easily build plenty of high rises but they are either insanely overpriced or extremely poor quality in London.
Data centres have big halls, usually no windows whatsoever, and are wide and flat so windows are only possible at the edges. They're even worse for conversion than office buildings.
Better to raze them and build apartment buildings.