Well, you've had a century of each so the styles have changed many times over the years...
Bristish cinematography has been very innovative and influential, especially the groundbreaking color work of Ozzie Morris & Jack Cardiff in the late 1940's / 1950's, and later David Watkin in the 1960's (there are too many other important DP's to list...)
But they were all stylistically different from each other. In general, European and U.K. cinematography tends to be more naturalistic, lighting "spaces not faces" so to speak (i.e. they light a room naturalistically and let actors move around in that light, rather than light for actor's faces first and then the background separately. But there are exceptions.)
But Jack Cardiff's work was more inspired by painters, particularly Van Gogh's use of color for example.