LONDON — Theresa May emerged on Monday as Britain’s next prime minister, becoming the second woman, after Margaret Thatcher, to hold that post. She faces multiple challenges as soon as she takes office, from negotiating the nation’s departure from the European Union to addressing deep social and political divisions to reviving a wounded 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.

Ms. May, until now Britain’s home secretary, is set to take over at a time of immense upheaval. 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 already taken a substantial hit from the exit vote, or “Brexit.”

A Conservative like Mrs. Thatcher, who governed from 1979 to 1990, Ms. May, 59, has won a reputation for steeliness in her tenure as home secretary. She has pledged to negotiate a deal to leave the European Union that would reassert Britain’s ability to control immigration, a central issue in the referendum on June 23 on whether to leave the bloc.

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

Ms. 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 for the “privileged few,” but for all.

The British referendum on June 23 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. Mr. 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 on Monday.

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

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

Ms. Leadsom had already faced accusations — which she denied — of embellishing her curriculum vitae, 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, while Ms. May is not.

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

The turn of events meant that Ms. May would become prime minister without a general election and without completing the campaign she and Ms. 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 Ms. May as a “bloody difficult woman,” noting that he had worked for another female politician of similar temperament, namely Mrs. Thatcher.

Far from being insulted, Ms. May used the comment to suggest that her negotiating stance with European officials would be as tough as that of Mrs. Thatcher, who 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 on Monday, Ms. May outlined an economic agenda unlike that of Mrs. Thatcher, calling for new mechanisms to curb executive pay and warning big multinational companies that they must pay their share of taxes.

Ms. May has also been compared to Angela Merkel, the chancellor of Germany who, like Ms. May, is the daughter of a clergyman and is known for her no-nonsense, methodical and pragmatic approach.

The post-referendum turmoil in Britain has also gripped the opposition Labour Party, prompting a challenge against its leader, Jeremy Corbyn, who has been accused of not campaigning hard enough against a British withdrawal from Europe. There were growing fears on Monday that the party might split because of the bitter dispute.

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, Ms. 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 E.U. There will be no attempts to rejoin it by the back door, no second referendum.”

But, while 52 percent of voters supported Brexit, they did so for differing reasons, including reasserting national sovereignty and opposition to migration from within the bloc, which guarantees the right to live and work within any member nation.

Ms. May is expected to take a tough stance on immigration in part because she is politically vulnerable on the issue. As home secretary, she failed to fulfill a Conservative pledge to control the number of people arriving in Britain. The numbers persistently exceeded targets and included arrivals of non-Europeans, which the government had the power to curb.

Ms. May has signaled her intention to restrict migration from Europe, even if doing so hurt Britain’s position in negotiating a new trade deal with the bloc. She will also be under pressure from big business to secure the best possible access to Europe’s single market, which normally entails accepting the principle of free movement of workers across national frontiers.

One important question for Ms. May is when to invoke Article 50 of the European Union’s governing treaty, which starts the withdrawal procedure. That effectively sets a two-year deadline for a deal to be struck. She has suggested that she intends to wait until her government has settled on its negotiating stance before invoking the article despite pressure from Europe’s leaders to act more quickly.

On Monday she said more about her wider political agenda in a speech that offered plans to address some of the economic and social inequalities evident in Britain. Those include populist resentments thought to have motivated many pro-Brexit voters in economically depressed areas outside London and in the affluent southeast of England.