All of Britain's messy politics go on display in one little by-election

In his blue suit and silvering hair, Mike Greene looks at home on the cathedral square in this English city where he grew up. Squint at his blue rosette as he shakes hands with voters and you might think this self-made millionaire and first-time candidate is stumping for the center-right Conservative Party, the party of Margaret Thatcher.

But the rosette Mr. Greene wears is turquoise, not blue, and he is standing for the Brexit Party, the newest disruptor in Britain’s fractious politics.

On Thursday, the party came within a whisker of taking Peterborough, a bellwether of national politics, from the opposition Labour Party. The margin of victory was 683 votes, and both Labour and Conservative saw double-digit falls in their share of the vote.

The by-election wasn’t the knockout blow that some had predicted after the Brexit Party – which arrived on the scene in February –