Boris Johnson’s last act as Britain’s foreign secretary was to gather a host of European dignitaries in London and fail to turn up.

As the group of ministers waited in a conference center in London’s docklands last summer, text messages flooded in: Mr. Johnson had just quit the British government in protest over its handling of Brexit. “Well, it’s all rather happening, isn’t it?” said Alan Duncan, a British foreign office minister who was playing host amid the diplomatic mess, according to people present. European officials...