Labour Party Brexit spokesman Keir Starmer tweeted: "I wonder what it is about the pending Tory leadership contest that made Boris change his mind?"
May earlier told her Conservative Party MPs: "I have heard very clearly the mood of the parliamentary party. I know there is a desire for a new approach, and new leadership, in the second phase of the Brexit negotiations, and I won't stand in the way of that".
"I am prepared to leave this job earlier than I intended in order to do what is right for our country and our party," she said.
Quitting after the deal passes would mean May would have achieved her objective of Brexit, but wouldn't then have to stay in control for the long negotiations with the EU over Britain's relationship with the bloc.
The BBC's political editor Laura Kuenssberg commented on Twitter: "May becomes another Tory leader whose time in office is cut short by party's anguish and division over Europe".
May has battled to keep the Conservative Party's pro-Brexit and pro-Remain factions together throughout the process. Her Government has also been reliant on support from the Northern Ireland DUP to maintain control.
Critics were quick to note that May didn't offer a specific exit date.
"If I were them, I'd want it in blood," Margaret Beckett, a Labour Party politician, told the BBC. "I've lost track of how many times she's promised she won't lead them into the next election and suddenly it's turned out that she might after all."
The Prime Minister's announcement is seen as key to getting dozens of hard Brexiter lawmakers, including Boris Johnson and Jacob Rees-Mogg, to back her Brexit deal.
After May's earlier promise of a tentative, slow-motion resignation, the Parliament was set to vote today on what kind of Brexit the fractious, quarrelling, paralysed MPs want.
Even with May's bold move, it is possible their answer will be a mush.
It is also possible that a way forward might emerge in a series of "indicative" votes for or against a "soft" or "hard" exit from the European Union.
The Parliament may signal whether MPs want a political and trade relationship with Europe like Norway has - or whether Britain should go it alone.
MPs could also push for a second referendum to give voters another say on how, or whether, they want to leave the EU.
Traditionally, the British Parliament has a brake but not a steering wheel. This week, though, MPs have seized control - for a couple days - of Britain's exit process.
The votes that begin today will be nonbinding. May has said she is willing to hear what the lawmakers have to say, while at the same time she worries the votes will stoke a constitutional crisis. May's Government opposed the voting scheme but is too weak to stop it.
The House of Commons has in the past three months said what it does not want. Lawmakers two times voted overwhelmingly against May's Brexit deal. They also said they do not want Britain to leave the continental trading bloc without a deal.
Speaker John Bercow chose eight of 16 proposals submitted to proceed to a debate and a vote.
Why 16 when you can have 17 options? asked the Twittersphere, which saw social media users on Wednesday coming up with amusing alternatives for how to break the deadlock, including "Bring in Judge Judy" and "Let the Queen decide."
Among the actual proposals brought forward is a Norway-style relationship with the EU. That would allow frictionless, tariff-free trade, but it might make Britain accept the free flow of European migrants - a major objection among those who voted to leave the EU.
The opposition Labour Party, meanwhile, is pushing an "alternative customs union" Brexit plan.
Other options include leaving the EU without a deal or canceling Brexit altogether.
Labour MP Hilary Benn told the BBC that he wanted the Prime Minister "to look at what we might be able to agree on and say, 'OK, I'm prepared to move.' That is what good leadership is. Unfortunately there hasn't been any since this began."
Labour leader Jeremy Corbyn told Parliament that May had a "very clear choice . . . either listen and change course - or go."
She is "failing to deliver Brexit because she can't build a consensus, is unable to compromise and unable to reunite the country," he said.
A recent YouGov poll found that 31 per cent of the public preferred May as prime minister, 19 per cent said Corbyn, and 46 per cent were not sure.
May has so far indicated that she is pressing on with the withdrawal deal she negotiated with EU leaders. She suggested that she could bring back her deal - dubbed MV3 - for a third time this week.
But Bercow issued a fresh warning.
The Commons Speaker said earlier this month that he would only allow another vote on the deal if it had been substantially changed since the last time it was rejected by MPs.
Bercow today warned the Government he still expected that "test of change" to be met as he also cautioned ministers against trying to find a procedural loophole to get around his ruling.
Downing Street had been targeting Saturday this week to potentially hold the third vote on the deal after a number of senior Tory Brexiteers appeared to soften their opposition to the Prime Minister's agreement.
But Bercow told the House of Commons: "I understand that the Government may be thinking of bringing meaningful vote three before the House either tomorrow or even Friday if the House opts to sit that day.
"Therefore in order that there should be no misunderstanding I wish to make clear that I do expect the Government to meet the test of change.
"They should not seek to circumvent my ruling by means of tabling either a notwithstanding motion or a paving motion.
"The table office has been instructed that no such motions shall be accepted."
During lively Prime Minister's Questions, May told a fellow Conservative lawmaker who has voted twice against her deal: "We can guarantee delivering on Brexit if this week he and others in this House support the deal."
On the other side of the English Channel, the Europeans wait and wait. Donald Tusk, President of the European Council, told legislators Wednesday in Brussels that the EU should be open to a long extension if Britain wanted to "rethink" its strategy.
"You cannot betray the six million people who signed a petition to revoke Article 50 - the 1 million people who marched for a people's vote or the increasing majority of people who want to remain in the European Union," Tusk said.
Tusk was referring to the "Cancel Brexit" petition hosted on Parliament's website, alongside the mass demonstration staged Saturday in London - probably the biggest public demonstration in Britain in a century.
Jacob Rees-Mogg, a Conservative MP and something of a power broker in these negotiations, said he would back May's deal as long as Northern Ireland's Democratic Unionists did. He tweeted that "half a loaf is better than no bread."
Explaining the U-turn, he told the BBC that he changed his mind because the government has backed away from his preferred option, leaving without a deal, and May's deal at least delivered Brexit.
Things may be confusing on the Conservative benches, but they aren't any clearer on the Labour ones.
Speaking on BBC Radio 4, Labour MP Peter Kyle spoke about his preferred Brexit option, that any deal passed should be put to a public vote. He said that during the "indicative" voting, Corbyn "will order" the party's MPs to vote for this.
Labour MP Barry Gardiner didn't appear to get that memo. He told the same radio programme that option was "not where our policy has been" and that "the Labour Party is not a remain party - we have accepted the result of the referendum." His name was soon trending on social media.
Like the Conservative Party, the Labour Party has been trying to walk a tightrope, appeasing both the "remainers" and "leavers" in their party. It did not go unnoticed at the massive on march Saturday - where an estimated 1 million took to the streets to demand a second referendum - that Labour sent its deputy leader to address the crowds but not its leader.
- NZ Herald, Washington Post, Daily Telegraph, AP
More to come
