LONDON — After an abrupt end to a dramatic leadership struggle, Theresa May, the home secretary, emerged Monday as Britain’s next prime minister, with responsibility for negotiating the nation’s withdrawal from the European Union, addressing deep political and social divisions, and managing a slumping economy.

Her victory came on another day of rapid developments in British politics and set in motion a process that officials said would put her in 10 Downing Street by Wednesday night, succeeding David Cameron as prime minister.

May, 59, would become the second woman to lead Britain, after Margaret Thatcher, who governed from 1979 to 1990.

She is set to take over at a time of immense upheaval for Britain. The nation must not only negotiate its withdrawal from the European Union, a process fraught with economic and political risks, but it must also hold itself together amid a renewed clamor from Scotland for independence.

Early signs are that Britain’s economy has taken a substantial hit from the exit vote.

A Conservative like Thatcher, May has won a reputation for steeliness in her tenure as home secretary. She has pledged to negotiate an EU deal that reasserted Britain’s ability to control immigration, a central issue in the vote on June 23 on whether to leave the bloc.

But unlike Thatcher, May is seen as a relative political moderate and on Monday she promised to address inequality, give workers greater representation on corporate boards, and limit tax cuts.

May said that she was “honored and humbled’’ to be chosen for the job, promised to get the best deal over Britain’s exit from the European Union, and vowed to create an economy that works not only for the “privileged few’’ but for all.

The British referendum plunged the European Union into crisis. It left Britain rudderless as the pound sank in value and both the governing and opposition parties engaged in fierce and acrimonious leadership battles.

Cameron, who had supported remaining in the European Union, said after the referendum that he would resign once the governing Conservatives selected a new leader, a process that he had expected to last until September but that was drastically accelerated Monday.

After making a brief statement describing his successor as “strong’’ and “competent,’’ Cameron was caught on a microphone humming as he returned to his office.

Cameron’s statement completed a day of high political drama in which May’s rival for the Conservative Party leadership, Andrea Leadsom, the energy minister, withdrew from the race. Leading party members quickly coalesced around May, insisting that the contest should not be reopened. A party committee agreed.

Leadsom had already faced accusations — which she denied — of embellishing her resume, but her campaign ran aground over the weekend after The Times of London published an interview in which she suggested that she was a better candidate because she is a mother, which May is not.

Surrounded by supportive lawmakers, Leadsom on Monday made no reference to that issue but said that she was quitting the contest and endorsing May and that she hoped to see her installed as prime minister “as soon as possible.’’

The turn of events meant that May would become prime minister without a general election and without completing the campaign she and Leadsom had been waging for the endorsement of the Conservative Party’s rank-and-file members.

Last week, the former Conservative Cabinet minister Kenneth Clarke described May as a “bloody difficult woman,’’ noting that he had worked for another female politician of similar temperament, namely Thatcher.

Far from being insulted, May used the comment to suggest that her negotiating stance with European officials would be as tough as that of Thatcher, who famously went into battle on the Continent, particularly over British financial contributions to the bloc.

“Ken Clarke might have found me to be a ‘bloody difficult woman.’ The next person to find that out will be Jean-Claude Juncker,’’ she told fellow Conservative lawmakers, referring to the president of the European Commission, the bloc’s executive, ITV reported.

But in a speech Monday, May outlined an economic agenda unlike that of Thatcher, calling for mechanisms to curb executive pay and telling big multinational companies they must pay their share of taxes.

Her top priority will be to frame a negotiating strategy for leaving the union, after the referendum in which she sided with those who wanted to remain.

On Monday, May insisted that “Brexit means Brexit,’’ as she sought to reassure right-wingers that she was committed to the policy, adding that “there will be no attempt to remain inside the EU. There will be no attempts to rejoin it by the back door, no second referendum.’’