Just medium-to-high density by
UK standards around existing larger train stations would do an immense amount to alleviate the UK housing crisis.
I used to live on a road with ~660 houses. The road, the houses, and the yards combined took up ca. 40,000 m^2. Near the local well-connected train station, a few new-builds in the 20~ floor range added more homes than that entire road took up in ~2,000 m^2.
On top of that, because of its location, the strain added on transport was far less - most people would be a short walking distance from both shops and commuter trains to the centre.
There are hundreds of major transit hubs like that in the UK where you could easily add do relatively modest hubs of higher-density housing and add a massive amount of capacity.
What's lacking is the political will to solve the massive problem of house owners who have been trained to see rising house prices as financially beneficial to them.
When most of those house owners are in an older demographic more likely to vote, it's a huge challenge to fix.