The whole commoner/royal thing is not very strictly defined in this ATL United States. Obviously, early on, several of the states had their own noble governments, but nobility has always been, in certain ways, inherently equal to the common people: legally, the King is an officer of the state, not the state itself. The American Crown is the executive branch of the government, but it is not supreme, de jure or de facto, over the American Legislature and the American Judiciary.And as a commoner "President Locklear" obviously has no relation to any of these royals?
Anyway, yeah, President Locklear would be essentially the prime minister of the United States, in a semi-presidential system. Though, as I see it, it's a system in which the Head of Government is generally dominant, but the Head of State is not totally declawed as in the RL U.K., due to the strict definitions of power in the Constitution that still delegate certain critical powers to the monarch, as it was written in the eighteenth century. And developed in a world without significant, stable and sovereign constitutional republics until the twentieth century.