LONDON—The woman charged with guiding the U.K. through its divorce from the European Union isn’t known for flashy political skills. Theresa May’s talks with EU leaders have been strained. She cuts a lonely figure at summits, standing alone as others chat amiably. In bilateral meetings, leaders have grown uncomfortable with her long silences and awkward manner.

What she is known for is determination and pragmatism, shown in how she emerged from the political chaos following the country’s 2016 Brexit referendum. She was hailed...