LONDON—A look at one recent week in Britain’s government illustrates how the country is drowning in Brexit.

On Thursday of that week, the U.K. Parliament debated turning a stretch of highway in southern England into a giant parking lot for trucks should ports get blocked by customs controls after Brexit. On Wednesday, six lengthy committee sessions were devoted to leaving the European Union. And on Monday, the British cabinet was briefed on how ships could be rented to carry goods to the U.K. if no exit deal with the EU was...