British House of Commons leader Andrea Leadsom has announced her resignation from the government over Prime Minister Theresa May's handling of the Brexit crisis.

The senior member of the Conservative Party said in a letter to Mrs May that she was quitting because "I no longer believe that our approach will deliver on the referendum result", which triggered Britain's efforts to leave the EU.

It is the 36th ministerial resignation under Mrs May’s premiership.

The beleaguered premier is in the last throes of a tumultuous rule focused all-but exclusively on guiding her fractured country out of the European Union in one piece.

But three overwhelming rejections by parliament of the terms she struck with the other 27 nations last year have forced Britain to miss the original 29 March departure date and plead for more time.

Mrs May is now paying the price for failing to deliver on the wishes of voters who chose by a narrow margin in 2016 to break from the EU.

Anxious members of Mrs May's party were meeting behind closed doors today to discuss changes to the rules that would let them vote no-confidence in her leadership in the days to come.

Mrs May has already promised to step down no matter the outcome of her fourth attempt to ram her version of Brexit through parliament in early June.

But even that sacrifice - and a package of sweeteners unveiled that included a chance for MPs to get a second Brexit referendum - failed to win hearts and minds.

"It's time for the prime minister to go," Ian Blackford of the pro-EU Scottish National Party told Mrs May as she tried to defend her latest proposals in parliament.

"Will she do it?"

Mrs May ignored the question and called the upcoming vote Britain's last chance to leave the EU with a negotiated deal that can avert economic chaos.

"There are people who tell me I have compromised too much in the package being put forward, others telling me I have not compromised enough in the package being put forward," she said.

"At some stage the House has to come together and we have to decide to go the distance together in order to deliver Brexit."

But things look to go only worse for Mrs May in the days and weeks to come.

Her Conservatives are set to get thumped in European Parliament elections on Thursday in which the brand new Brexit Party of anti-EU populist Nigel Farage is running away with the polls.

The European elections are being interpreted in Britain as a referendum on both Brexit and Mrs May's ability to get the job done. They make grim reading for the government team.

A YouGov survey showed Mr Farage's Brexit Party claiming 37% support.

The pro-EU group of Liberal Democrats was second on 19%. The main opposition Labour Party was on 13% and Mrs May's Conservatives were lagging in fifth place with just 7%.

Mrs May is still hoping to stay in power long enough to somehow win parliament's approval of the EU divorce terms before its summer recess begins on 20 July.

This would let the country leave at the end of that month - as long as MPs reject a second referendum.

Otherwise the process could be delayed until 31 October - the deadline set by the EU - or even later if its leaders grant Britain another postponement.

But pressure within both Mrs May's government and party is building for her to go now so that a new leader can rescue the process before Britain crashes out without a deal.