The world is looking at Britain and asking: What on Earth just happened? Those who run Britain are asking the same question.

Never has there been a greater coalition of the establishment than that assembled by Prime Minister David Cameron for his referendum campaign to keep the U.K. in the European Union. There was almost every Westminster party leader, most of their troops and almost every trade union and employers’ federation. There were retired spy chiefs, historians, football clubs, national treasures like Stephen Hawking...