The U.K. has spent six months debating where it wants to go. But now, with just three months until Prime Minister Theresa May intends to formally start the Brexit process, it is grappling with a new question: How to get there?

In any negotiation, process is power—and there is growing anxiety in the British government that it may be forced to accept a Brexit process that will give the EU the upper hand. Some officials privately warn that under those circumstances, the U.K. could even abandon the process altogether, taking...