BRUSSELS—Boris Johnson’s general election victory, and the likely departure of Britain from the European Union next month, will bring relief to most European governments: Now they can focus on other pressing issues facing the bloc.

Yet Brexit was a rare point of unity for the remaining 27 members and life beyond it could expose divisions among them. It isn’t clear, for example, how cohesive those left in the bloc can be as they confront issues after Britain’s departure—including negotiating new trade relations with the U.K.

...