LONDON—British Prime Minister Boris Johnson, fighting a serious coronavirus infection, was admitted to the intensive-care unit of a London hospital Monday evening, leaving his country’s leadership in question at a moment of deep crisis as the virus’s spread accelerates.

As Mr. Johnson’s condition worsened, he deputized his foreign secretary, Dominic Raab, to run the government, a spokesman said. The 55-year-old premier was conscious when he was wheeled into the intensive-care unit of a central London hospital at around 7 p.m....