A Filipino cardinal known for riding public transit and making YouTube videos of himself singing and playing guitar is one of the leading contenders to follow Pope Francis, whose poor health is raising questions over his tenure as head of the Catholic church. Other prospects include a more daring liberal, Pietro Cardinal Parolin, the Vatican secretary of state; a Hungarian, Peter Cardinal Erdo, who is frontrunner in the conservative camp; and a compromise candidate, Matteo Cardinal Zuppi, the archbishop of Bologna. These names are moving to the van as Francis’s health is failing visibly. Last week he canceled a trip...