LONDON—British Prime Minister David Cameron, fresh from narrowly dodging the breakup of the U.K. after Scottish voters rejected independence, kicked off a new wave of political uncertainty with a surprise proposal to fundamentally reshape the centuries-old union.

Mr. Cameron's gambit, announced shortly after the final ballots were tallied Friday, calls for transferring more powers not just to Scotland—as the British leader had already pledged—but also to England, Wales and Northern Ireland.

The push to give the four parts of the U.K. greater independence could mark a reordering of how issues such as tax, welfare and public spending in the U.K. are governed. Mr. Cameron said he wants the new laws drafted as soon as January, setting the stage for a contentious political debate ahead of general elections next May. Though all three parties support decentralization, there are considerable differences among them on which powers should be shifted. Some lawmakers within Mr. Cameron's own Conservative Party also were disgruntled over his handling of the referendum and last-ditch pledge to promise more authority to Scotland.

Throughout the union's history, such authority has long been centralized in London. Scotland, Wales and Northern Ireland have varying degrees of power, though little control over economic policy, including taxation. The changes Mr. Cameron has promised would be a step closer to the federal structure of the U.S., where there is a constitutionally entrenched division of authority between the central government and the states.

"Just as the people of Scotland will have more power over their affairs, so it follows that the people of England, Wales and Northern Ireland must have a bigger say over theirs," Mr. Cameron said outside Downing Street on Friday. "The rights of these voters need to be respected, preserved, and enhanced," he said.

Mr. Cameron's proposal wasn't the only surprise in the wake of defeat for Scottish nationalists. Scotland's First Minister Alex Salmond —who had spearheaded the pro-independence campaign—announced Friday his plans to stand down as leader of the Scottish National Party in November, and as First Minister once his party had chosen a new leader. Still, he said the fact that nearly 1.6 million people had voted to exit the union—roughly 45% of the total vote—was in itself a victory for the pro-independence camp.

Mr. Salmond, who had forced Mr. Cameron's hand to hold a referendum, has moved from being a fringe player in U.K. politics to becoming a figure whose political acumen helped bring the country to the brink of separation He secured an unexpected win for his party in 2011, giving Mr. Salmond an outright majority at the Scottish Parliament. Almost straight away, he announced plans for a referendum on independence.

Mr. Cameron's surprise move to offer more powers to other parts of the U.K. in addition to Scotland appears to be an attempt to appease members of his Conservative Party, analysts said. Many Conservatives have been disgruntled that Scottish voters were being offered greater powers in the run-up to the referendum but that their own constituents on the other side of the border in England hadn't.

The prime minister, who faces a general election in May, is under pressure to rally his center-right party. He failed to win the 2010 general election and was forced to form a coalition government with the centrist Liberal Democrats. Since then, he has faced repeated rebellions by his backbench politicians who have been unhappy with his leadership on issues ranging from the U.K.'s membership in the European Union to same-sex marriage.

Pressure from his party over Europe has already forced him to promise a referendum over whether the U.K. should leave the EU—a vote he has promised to hold if he wins the May election.

The U.K. has had a long-running debate about whether lawmakers representing constituencies outside England should have a say on affairs that only affect the English, while those in England have no say on similar matters that affect Scotland, Wales, and Northern Ireland. Unlike Scotland, Wales and Northern Ireland, England doesn't already have some form of devolved government or its own parliament.

Mr. Cameron said he has put former Foreign Secretary William Hague in charge of drawing up plans and that a cabinet committee would be set up immediately. He added that he hoped other political parties would contribute.

The Labour Party, the main opposition party, as well as Mr. Cameron's junior coalition partner, the Liberal Democrats, voiced support for the move.

Some Conservative lawmakers said they welcomed the move but awaited more detail.

John Redwood, a former Secretary of State for Wales in former Prime Minister John Major's cabinet whose constituency is in England, said the shift in powers were appealing, especially in England, given that Conservative law makers hold a majority of English seats in Parliament.

"We just want to get on and do conservative things in England in accordance with our majority," Mr. Redwood said.

Mr. Cameron faces stiff hurdles in putting his plan into action, including on the thorny issue of whether lawmakers representing constituencies outside England should have a say on some English affairs.

The Labour Party would likely dissent on this point because without votes from Scottish, Welsh, and Northern Irish politicians, Labour would struggle to win majorities on some legislation at Parliament.

"It seems extremely unlikely you will get a consensus on the English side of this in the timetable framed," said John Curtice, professor of politics at the University of Strathclyde. "If he can't sort this out for the English, he won't be able to do it for the Scottish."