LONDON — Theresa May, the front-runner to become Britain’s next prime minister, revealed in 2013 that she had been given a diagnosis of Type 1 diabetes, a condition that requires daily insulin injections.

Asked later how she felt about the diagnosis, she said her approach to it was the same as toward everything in her life: “Just get on and deal with it.”

That kind of steeliness has propelled her to center stage in the aftermath of Britain’s vote to leave the European Union and the feuding that erupted in the Conservative Party over who would succeed David Cameron, the current prime minister, who said he would resign by the fall.

Conservative lawmakers began voting on Tuesday to winnow the five contenders to two whose names will go to party members for the final say. In the first ballot, Ms. May led with 165 votes; Andrea Leadsom was second with 66. Liam Fox received the fewest votes and was eliminated, and Stephen Crabb, who came in fourth, exited the race, lending his support to Ms. May. The balloting process is scheduled to be completed next week.