Donald Trump’s overnight transformation from a candidate scorned by the Republican establishment to the party’s presumptive presidential nominee will ramp up pressure on GOP leaders and elected officials to fall in line behind him.

Until now, support for Mr. Trump’s campaign from party leaders hasn’t grown nearly as rapidly as have his popular-vote totals. Only 12 of the 300 Republicans serving in the House and Senate, and three of the 31 Republican governors, have endorsed the New York businessman, despite a monthlong winning...