Tens of thousands of Americans (not to mention Canadians and other Allied service personnel) did indeed find partners in the UK (and across the rest of the world):

"The world war of the 1940s, fought around the globe, produced marriages by the scores of thousands, and did so in every region where American soldiers were posted. [...] The number of GI marriages in World War II, though impossible to determine precisely, was over 125,000, with more than eighteen countries producing a thousand marriages or more." (Entangling Alliances: Foreign War Brides and American Soldiers in the Twentieth Century, Susan Zeiger)

Few remained in the UK, though. Aside from the gender norms of the time expecting, in general, wives to follow their husbands, post-war Britain was hardly a paradise. The experience of GIs during the war tended to be of "perpetual damp, shocking plumbing, weak beer, boring music, worse food" (Britain's War: A New World, 1942-1947, Daniel Todman), even with the end of the war rationing persisted (for almost ten years). Where relationships did last, by no means guaranteed, most spouses emigrated to the United States or Canada with their husband (or wife - members of the Women's Army Corps (WAC) also met partners overseas; the 1949 screwball comedy I Was a Male War Bride was, loosely, based on fact). This was facilitated by the War Brides Act of 1945 that allowed spouses and children to enter the US without being subject to the quotas of the Immigration Act of 1924.

In the 14 months from VE Day to June 1946, 38,723 dependents were shipped from the United Kingdom to the United States by the Army, 28,299 adults and 10,424 children. The Ships of Operation War Bride: Embarking War Brides and Dependents to America, Canada, Australia and New Zealand from the Maritime Archaeology Trust has some more details. There do not appear to be specific figures for how many GIs remained; "We do not know, for instance, how many couples elected to stay in Britain, but anecdotal evidence suggests that the figure was small." (Rich Relations: The American Occupation of Britain, 1942-1945, David Reynolds).