SriLanka, July 20 -- Britain has become accustomed to shuffling through prime ministers at a rapid pace and is now poised to have its seventh leader in a decade. But the transition still requires a few steps.

Last month, Prime Minister Keir Starmer, amid a mutiny from within his party, announced his plans to resign. That made way for Andy Burnham, the former mayor of Greater Manchester, who became eligible by winning a special election for a seat in Parliament.

Under Britain's parliamentary system, voters don't elect an individual to serve as prime minister but rather cast their ballots for a party to govern. It is the party in power, in this case Labour, that gets to decide who is at the helm.

Mr. Burnham made an uncontested bid for l...