LONDON— Theresa May is set to become the U.K.’s prime minister Wednesday, inheriting a long list of trade and political challenges in the divided nation that voted last month to leave the European Union.

Ms. May, the home secretary, will assume her new role at a time of political and economic upheaval and will have the contentious task of negotiating Britain’s withdrawal from the EU.

She is set to formally become prime minister at the end of a traditional sequence that kicks off Wednesday afternoon, in the hours after departing leader David Cameron attends his final Prime Minister’s Questions, an often raucous weekly session where lawmakers grill the leader on issues facing the country.

Mr. Cameron will head to Buckingham Palace, tender his resignation to the queen and recommend Ms. May as his successor.

Shortly after, Ms. May will arrive at the palace. She will be photographed with the queen in the private room where the monarch and prime minister hold regular meetings, and formally become the U.K.’s second woman to hold the role, after Margaret Thatcher. Ms. May will be the 13th prime minister that the queen has worked with during her 64-year reign.

Ms. May’s appointment will cap three weeks of turmoil after the Brexit vote prompted Mr. Cameron to announce his intention to resign.

Under British constitutional rules, the choice of who leads the country falls to the party with enough seats in parliament to form a government—in this case the Conservatives, who secured a majority in last year’s general election. The party is free to choose who it wants to lead the country any time until the next mandated national vote takes place in 2020.

After Mr. Cameron said he would step down, a handful of Conservative lawmakers jostled to replace him. Weeks of political positioning winnowed out some would-be candidates, and voting by lawmakers further narrowed the field to the anti-Brexit Ms. May and the pro-Brexit Andrea Leadsom. The race ended unexpectedly Monday, after Ms. Leadsom—under fire for telling an interviewer that she was more qualified to lead the nation because Ms. May has no children—dropped out. She said a drawn-out leadership contest wasn’t in Britain’s interest.

As home secretary, Ms. May had a brief spanning law enforcement, immigration and domestic security. She was one of the country’s longest-ever serving home secretaries and gained a reputation for firmness, facing off against police over funding and oversight and pushing immigration limits.

Ms. May will have to oversee a complicated divorce from the EU that will likely take years. She will also have to heal a divided party, which was torn apart during the referendum campaign and further split during the leadership contest.

One of her first major decisions will be when to formally trigger Britain’s exit from the bloc under Article 50 of the EU’s Lisbon Treaty. Once she gives official notice of the withdrawal, the U.K. will have two years to negotiate terms of separation.

Ms. May will soon choose a new cabinet, which will likely include prominent figures from both sides of the EU referendum debate as a way to unify the party. One major question is whether George Osborne, Britain’s Treasury chief, will be able to hold on to his post. She will also have to choose a politician to lead the government’s Brexit unit, which will focus on the U.K.’s separation from the EU.