Very few house here in the UK have basements and I'm not aware of any new houses being built with basements, they seem to be the domain of many of the older very small houses in inner city areas. (unless you know differently)
In Brighton and Hove (where I lived for much of my childhood) almost all the pre-WWII housing stock has basements (or basement flats/rooms), which tend to be very damp as the water from the chalk Downs percolates down to the shoreline through everything. Conversely, almost nothing built there after the war has a basement unless it was a re-build over an existing site that was already excavated for it (when presumably it would have been more expensive to fill the hole than slap in a basement).
I suspect that pre-war/post-war dividing line is true for a lot of urban Britain as it coincides with the widespread adoption of domestic refrigeration (somewhere cool for food storage) and the decline in use of coal for domestic heating (coal cellar, obviously). However, that's probably only true for what, for want of a better phrase, I'm going to call non-working class homes. Tenements and terrace housing, of whatever age, typically don't have cellars in most towns. However, as I've implied, that's not true for Brighton where much of the terraced housing stock does seem to have a basement/sub-street level, but it's usually originally divided up as typical living quarters not storage.