The U.S. continues to lay claim to more elite research universities than any other country in the world, but that dominance is beginning to fray.

Oxford and Cambridge, the intellectual one-two punch of the U.K., took the first and second spots in the 2018 Times Higher Education World University Rankings. Their showing marked the first year schools outside the U.S. seized the two top positions in the 14-year history of the list.

The...