LONDON—The U.K. faces the prospect of prolonged political uncertainty after an inconclusive election cast doubt on Prime Minister Theresa May’s ability to stay in office or govern effectively.

A weekend of drama cost Mrs. May her two closest aides as criticism mounted over the prime minister’s missteps in an election she had hoped would strengthen her parliamentary authority before looming Brexit talks with Brussels.

Her Conservative Party now is trying to form a minority administration propped up by Northern Irish lawmakers.

Senior Conservative Party figures called for a more collegial approach to government after her tightknit inner circle failed to deliver an expected victory.

In a sign of confusion at the very top of government, Mrs. May’s office late Saturday had to backtrack on a statement that it had reached a deal with Northern Ireland’s Democratic Unionist Party over forming a loose alliance to support a new government. The DUP said instead that talks centered on a so-called confidence-and-supply arrangement, a weaker and potentially more unstable partnership than a formal coalition.

“We’re ready to roll,” Irish Foreign Minister Charlie Flanagan said on Sunday in an interview with U.K. broadcaster ITV.

Yet in Brussels, European officials preparing for talks on the timeline of the negotiations with their U.K. counterparts were wondering how meaningful talks can be. Mrs. May’s failure to win an outright majority in a national election on Thursday has cast doubt on her future as prime minister.

Boris Johnson, the flamboyant, pro-Brexit foreign secretary beloved by party activists, on Saturday dismissed as “tripe” newspaper reports that he was already planning a bid to unseat Mrs. May. A survey of more than 1,000 U.K. adults by polling firm Survation published Sunday found 49% thought Mrs. May should quit.

Jeremy Corbyn, leader of the main opposition Labour Party, said he is ready to fight another election and expects one to be called this year or in early 2018.

In broadcast interviews on Sunday, senior Conservative lawmakers said now isn’t the right time for a leadership challenge, given the added uncertainty it would cause as Brexit talks get under way.

“The last thing we need is further political upheaval,” said Transport Secretary Chris Grayling, an ally of Mrs. May.

But some said such a contest may need to happen soon. Nicky Morgan, a former education secretary who has clashed with Mrs. May, said she didn’t think the prime minister should lead the party into another election and that a leadership contest should be considered this summer, or ahead of the party’s annual conference in October.

“I think Theresa May is ultimately going to take responsibility,” she said, referring to the election result. Mrs. Morgan said the party needs a proper contest to select a new leader rather than “a coronation.”

Defense Secretary Michael Fallon on Sunday said that, meanwhile, Mrs. May’s cabinet expects to have a greater say in government following the resignation of her two top aides, Nick Timothy and Fiona Hill. The two had played a central role in driving government policy and strategy and oversaw the botched election.

“We are going to see I hope much more collective decision-making in government,” he said in an interview with the British Broadcasting Corp.

George Osborne, who served as Treasury chief in the administration of Conservative former Prime Minister David Cameron, was highly critical of Mrs. May. “Theresa May is a dead woman walking—it’s just how long she’s going to remain on death row,” he said on the BBC.

As well as calling into question Mrs. May’s future, Thursday’s election raises doubts about the party’s ability to deliver on its legislative platform.

The DUP’s 10 seats in Parliament are enough to give the Conservatives, with 318, a slender majority in the 650-seat assembly. But the deal being discussed between the two sides falls short of a formal pact that would allow the Conservatives to rely on DUP support on every vote. That raises the prospect that Parliament could defeat or amend the Conservatives’ plans on everything from Brexit to welfare and education.

A confidence-and-supply arrangement means the DUP at a minimum would pledge to back the government in any future no-confidence motions in Parliament and to support its tax-and-spending plans, delivered in twice-yearly budgets debated by lawmakers. Governments in the U.K. must win such votes to stay in power. It isn’t yet clear whether the arrangement will extend to other areas of policy as negotiations are continuing.

A big question mark hangs over the government’s Brexit strategy. The DUP says its priority is in preventing Brexit from causing any disruption to trade with EU member Ireland, a stance analysts say is incompatible with Mrs. May’s pre-election position that she was prepared to walk away from talks without a deal.

Mr. Corbyn said Conservative plans for a bill to alter or scrap EU legislation are probably also in tatters in the absence of the parliamentary majority needed to swiftly enact such a huge undertaking. He said his party plans to present its own alternative plan for government when Parliament reconvenes on June 19.

Mrs. May on Sunday announced fresh appointments to her cabinet. She had been expected to make more-dramatic changes after the election but almost all senior officials remained in place, underscoring her limited room to maneuver. Damian Green, an ally, was named first secretary of state, her de facto deputy. Michael Gove, a leader of last year’s Brexit campaign, was named environment secretary.