Pro-EU supporters outside the British Parliament buildings in London on Dec. 12, 2018. Mike Newberg | CNBC

British Prime Minister Theresa May has promised U.K. lawmakers a vote on delaying the U.K.'s exit from the European Union, if her Brexit divorce agreement is rejected again next month. May made a statement to MPs Tuesday afternoon, saying they will be given a chance to vote on a no-deal Brexit or delaying the U.K.'s departure. These votes will only come if they reject her withdrawal deal at a vote on March 12. It means the U.K. will only leave the EU on March 29 without a deal if lawmakers vote for that option. It therefore reduces the possibility of the U.K. crashing out of the bloc, but does introduce a new layer of uncertainty and complexity into the proceedings. Earlier on Tuesday, May held a meeting with her cabinet following media reports that she was "ready to rule out" a no-deal departure from the EU on March 29. British newspaper The Daily Mail reported early Tuesday that "a group of 23 dissidents met secretly at the (House of) Commons last night to discuss how to stop Britain leaving the EU without an agreement on March 29, with as many as 15 said to be ready to resign." Three of the ministers cited by the newspaper said they are prepared to back a move by MPs – in key parliamentary votes on Brexit on Wednesday – to force the prime minister to seek a delay to the U.K.'s departure if her parliament has not approved a deal by March 13 - a day after parliament is due to have a "meaningful vote" on whether or not to approve May's Brexit deal. WATCH: Brexit explained: What you need to know about the UK's big gamble

Labour demands

May is not only facing ultimatums from members of her own Cabinet but from the opposition Labour party too, ahead of parliamentary votes on "amendments" to May's Brexit deal taking place Wednesday — and which could determine the direction Brexit takes next. Labour leader Jeremy Corbyn made the shock announcement Monday evening that the party would back an another Brexit referendum if it's own proposals for what form Brexit should take are rejected. Corbyn said Labour would "put forward or support an amendment in favor of a public vote" to prevent what he called "a damaging Tory Brexit being forced upon the country," he said in a statement released by Labour. He also said that the party would back an amendment, being proposed by Labour's Yvette Cooper and Conservative Oliver Letwin, that would force the prime minister to delay Britain's departure from the EU if parliament has not approved a deal by mid-March, taking a 'no deal' departure on March 29 off the table. This amendment is widely seen as allowing parliament to take control of Brexit as it gives MPs a say on whether the U.K. can leave the EU without a deal. Lastly, Corbyn said Labour will again be tabling an amendment in favor of its own alternative (and already proposed) Brexit plan for a "comprehensive customs union" and "close alignment" with the single market. Amendments are expected to be tabled by a handful of members of parliament (MPs) and parties and are essentially proposals on what they think should happen next in the Brexit process. It gives MPs the chance to influence government policy and if enough of them approve or reject an amendment, can essentially force the government's hand on the course of Brexit. Experts say parliament is getting jittery about the direction Brexit is taking just four weeks ahead of the departure date and a "meaningful vote" (parliament's vote on May's Brexit deal) on March 12. "Although May has promised MPs another vote on the deal by 12 March, her move has failed to halt the momentum towards the (House of) Commons seizing the initiative to rule out a no-deal exit," Mujtaba Rahman, managing director of Europe at Eurasia Group, said in a note Monday. "Many Tory MPs who have previously given May the benefit of the doubt until now are increasingly worried about a cliff-edge departure and ready to allow Parliament to seize control of the Brexit process," he noted, saying that there was a "very real prospect" that MPs will support the amendment proposed by Cooper and Letwin to force May to seek an extension of the deadline to 'Article 50' (the EU departure process). "May might yet be forced to swallow a referendum to keep her deal alive, despite her strong opposition to the idea," Rahman noted, adding that as time runs short, even the most euroskeptic MPS are keen to avoid a "no-deal" Brexit.

No no-deal