The UK is already fairly urbanized. Basically everyone lives in or around a city. You could densify the cities if the residents actually want that.
What's being discussed here is not that though, it's the problem that everyone seems to want to go to alpha cities like London/SF/NYC and treats elsewhere as "inferior".
You can't just stuff the entire population of the UK in to London because it would be a hellscape regardless of how much you improve infrastructure.
Few seem to want to live in high density housing beyond their 30s. As soon as families are being thought about thoughts turn to having safe places to play, quality of air and schools, thoughts of the logistics of pram, a couple of children, nappies and bottles being manoeuvred up and down 10 floors. It's bad enough doing that when the car is on the street in front of your house.
That's the point people are trying, to the extent their finances allow, de-aggregating themselves to a house in the suburbs or somewhere pleasant and rural.
Sure the schools may be nearer in a city, but they often come with a poor reputation, or the nearby ones are full so you have to travel just as far anyway. Air quality and neighbours aren't a problem of bad architecture either.
Perhaps if we went back and levelled the place and rebuilt with some of the values of pre-war times we could create high density housing that worked. We'd need local shops, services, transport hubs and schools. The very things that have been consolidating away from local hubs throughout the past 50 years.
Walk about in Central London on a weekday and tell me if you doubled the population that'd be fine and dandy.