On Friday, senior transactional partners at many elite U.S. firms were working the phones nonstop to advise clients on the legal ramifications of Britain’s stunning vote Thursday to pull out of the European Union.

But behind the frenetic client counseling, several lawyers said that they were bracing for a possible two-year drought in their most highly leveraged, profitable business: advising on cross-border M&A transactions in Europe and the U.K.