Officials at the U.K.’s central bank are pushing lenders to sharpen up their contingency plans as their branches in Britain could lose the right to operate after March 2019—a situation that would further jeopardise London’s position as Europe’s key financial centre.

Neil Hodge is a freelance business journalist and photographer based in Nottingham, United Kingdom. He writes on insurance and risk management, corporate governance, internal audit, compliance, and legal...