Keep calm and carry on? Brexit fatigue is permeating Britain as the final moves in the long-running saga create confusion and unease about how it will all end up.

EU leaders on Thursday agreed to extend the Brexit deadline triggered two years ago, leaving the final date of Britain's withdrawal in doubt.

A British government source earlier this week said it was like witnessing "the last days of Rome" as Prime Minister Theresa May failed to secure Brexit by the original scheduled date of March 29.

The Conservative leader on Thursday persuaded EU leaders to delay Brexit until May 22 if next week - at the third time of asking - they can agree to the divorce deal struck between London and Brussels, and April 12 or a later date if they cannot.

Even the normally reserved May snapped on Wednesday, voicing her frustration at MPs' inability to get it "over and done with".

European leaders and analysts believe she still faces a mountain to climb in order to get MPs to back the withdrawal agreement.

"It's possible that she will convert enough of the dissenters to enable her vote to pass on the fourth or fifth attempt," Professor Iain Begg, from the European Institute at the London School of Economics university, told AFP.

"But the current indications are no: there are too many ERG hardliners who will never accept her deal," he said, referring to the European Research Group of hardcore Conservative Brexiteers.

What do MPs want?

Consensus amongst British lawmakers seems as distant as ever, even as they prepare for the possibility of a series of votes next week aimed at finding a Brexit strategy that commands a majority in parliament.

Nearly three years after the British electorate voted to leave the EU in a seismic 2016 referendum, their MPs - the vast majority of whom backed staying in the bloc - are still torn apart on how to do it.

And some don't want to do it at all.

"They have rejected every proposal to make Brexit a reality," Vernon Bogdanor, research professor at the Centre for British Politics and Government at King's College London university, wrote in The Guardian.

They have twice heavily voted against the withdrawal agreement struck between May and the 27 other EU leaders.

They have voted against a no-deal exit, against a second referendum, and against the main opposition Labour party's demand to stay in the customs union.

"In one sense, what Theresa May is saying is correct: parliament knows what it is against but it has to articulate what it's for," said Begg.

And if it can't, MPs might test the weary public's patience to the limits, with Brexit overtaking almost all aspects of British public life.

Demonstrations demanding Brexit be implemented or voted on again are taking place over the coming days, while a petition calling for Britain to stay in the EU has topped more than three million signatures.

Liberal newspaper The Guardian said Brexit had revealed a country that was "incoherent, entitled, incapable of compromise... wholly unrealistic and startlingly ignorant" about the European Union.

'I hate them all'

Journalists are not spared Brexit fatigue.

"I sat on the train heading into Westminster one morning this week and felt I could cry," wrote Matt Chorley, editor of The Times newspaper's Red Box political website.

"Let me tell you which people in politics I can't stand: everyone.

"Nobody has said anything new, interesting or remotely helpful in these debates for about two years.

"It's all so bloody depressing. I hate them all."

Even the government is not immune.

"This could be all over in two months, and we could see our families again," one government source said Thursday.

The "constitutional crisis", as the government's chief legal advisor describes it, stems from May's lack of a parliamentary majority.

She is therefore at the mercy of the various factions within her centre-right party, divided between eurosceptics and europhiles, and of the right-wing Northern Irish allies that the Conservatives rely upon for support.

After a meeting of the party's 1922 Committee of backbenchers, one Conservative MP told AFP: "There's no confidence, no leadership - it's a mess".