Boris Johnson’s surprising announcement on Thursday that he would not seek the leadership of the British Conservative Party — and by extension, succeed David Cameron as prime minister — leaves five candidates for the job. In a series of votes, Conservative members of Parliament will whittle down the list to two finalists, and the party’s roughly 150,000 members will choose between them. The process is expected to be completed by Sept. 9. The Conservative Party, which won a majority in Parliament last year, has a five-year mandate that lasts until 2020. But given the turmoil after voters decided that Britain...