In the uk, whichever political party gets a majority gets to choose who they want to be PM, as long as that person is currently or is about to become a minister of parliament (MP). After they pick who they want, they go visit the king and everyone allows the king to pretend that he picked the PM to run the government. The king technically has the power to chose whoever they want, but if they ever tried to use that power, parliament would change the constitution to take the power away (just like every power the king has on paper but is not allowed to use in reality)