I think there are many influences which all contributed to the development of 20th century British literature. I’m not sure that there’s one influence that stands out more than the rest. Two that come to mind immediately are James Joyce and Virginia Woolf.
Although James Joyce is Irish, he is from the region and stands out among Modernist prose writers and might be considered the most innovative writer of the 20th century. He exemplified a lot of the Modernist tendencies including intertextuality, free verse or stream of consciousness and played with language to such an extent that his work characterized ambiguity in form and content. This would influence subsequent Modernist and Postmodernist writers because of the boundaries he broke with form and also because of the play of language which would influence the ambiguity of authors known for Existential, Absurdist, or even Post-structural themes.
The Modernist school had an influence on Britain, America and many other countries. Other authors and artists of this school include T.S. Elliot, Pablo Picasso, Bertolt Brecht, Ezra Pound, George Orwell, Salvador Dali and Aldous Huxley. Some of these artists and authors are not British but their influence certainly reached Britain.
Virginia Woolf’ work, in style and content, had a large influence on British literature in the 20th century and on literature and feminism in general. For example, her novel Orlando explored gender roles, the concept of time and how history constitutes individuals. It was a trans-gender novel and thus broke boundaries in cultural roles and boundaries of genre, mixing non-fiction with fiction.