Some people did, if they had the money. Children were evacuated and dispersed. However, as always, people have to live within commuting distance of their work. And pretty much everyone was either conscripted, recruited into the manufacturing or logistic side of the war effort, or in one of the "exempt" professions keeping the country operating. My grandfather was one of those as a railway signalman.

I can't remember whether ration cards were location-restricted as well, or whether there were bureaucratic obstacles to freedom of movement.