I have been meeting with law firm leaders at U.K. firms in recent weeks. The mass exodus from Greater China by U.S. firms over the past 18 months, which dominated the headlines, meant all the attention had focused on the U.S. play in Asia. The role of U.K. firms had gone largely ignored.

So, it’s time to ask: Will U.K. firms regain market share and the upper hand now that so many of their U.S. rivals have up and left Greater China?