London -- British Prime Minister Theresa May was shut away on Thursday with some of her allies -- a shrinking group -- as she mulled whether to try for a fourth time to get Parliament to approve her Brexit plan, or to just pack it in and resign. Her fate was hanging in the balance as a frustrated British public went to the polls to vote in elections for the European Parliament that many had hoped never to take part in.

Already Britain's lawmakers have rejected three slightly varied versions of May's proposed divorce deal with the European Union. May was dealt another major blow on Wednesday when yet another of her Cabinet ministers, Andrea Leadsom, resigned, saying she couldn't support the latest iteration of the EU withdrawal bill.

May had been planning to introduce the proposal to Parliament by the end of this week, hoping for another vote. That plan has been scuttled, and it was unclear when or if British legislators would vote on May's forth draft.

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Leadsom said she couldn't back May's plan because it failed to "deliver on the referendum result" of 2016, when a majority of British voters opted to leave the EU, which Britain helped to found more than seven decades ago.

Thus far May has seen about 50 members of her government, including 12 cabinet members, resign. At least 33 of those people cited her handling of Brexit as their reason for leaving.

Britain's last EU election?

With the stalemate in Parliament, Britain's originally scheduled EU departure date of March 29 was extended, first to April 12, and now to October 31 which, as CBS News contributor Simon Bates notes, is Halloween.

As Bates reported in his "London Calling" segment this week, that has left Britain to hold elections that never should have happened: All European Union countries have been voting this week to send legislators to the European Parliament.

Technically, Britain is still a member, so by law it has to take part, said Bates, even though the U.K. is slated to leave the EU in five months.

Campaigning in Britain has been rather strange, Bates noted, with many people using the EU election as a protest vote. The polling ahead of the vote showed support for May's Conservative Party had all but collapsed.

Way out in front in the polls, was the newly formed Brexit Party, which has just one policy in mind: to get Britain out of the EU as soon as possible. In spite of its expected success in the EU parliamentary elections, however, public reaction to the party has highlighted the deep divisions within the U.K. over the decision to leave the EU.

Brexit Party leader Nigel Farage was doused with a well-aimed milkshake earlier this week, and then refused to leave his campaign bus at a subsequent stop as milkshake-wielding detractors loomed outside.

When the EU election results are announced Sunday night, all eyes will turn back to May. She's already said she'll resign once Brexit is agreed, but if the results are as dire as predicted for the Conservatives, her angry party might come after her and hold a no-confidence vote, which could force her to step aside.

If May does quit as party leader, she will most likely remain in the prime minister's office on a temporary, caretaker basis for a handful of weeks. That will give her party time to elect a replacement to occupy 10 Downing Street.

In spite of the turmoil surrounding her fate, May's spokesman insisted on Thursday that she would still be the sitting British Prime Minister when President Trump visits Britain for an official three-day state visit starting on June 3.

It could be close.