Their departure followed that of Foreign Secretary Boris Johnson, Brexit Secretary David Davis and Davis' number two, Brexit Minister Steve Baker. Meeting Conservative lawmakers on Monday, May was cheered and applauded by many, having warned that squabbling could pave the way for socialist opposition leader Jeremy Corbyn to take power. A year ago reports emerged that Davis said he meant to "retire" in 2019 and leave the transitional period of Britain's exit from the European Union to Boris Johnson, something later dismissed as a "lighthearted remark".
May named one of her most loyal ministers, Jeremy Hunt, to replace Johnson in the job of Britain's top diplomat.
They were announced less than an hour before May was due to face the press alongside German Chancellor Angela Merkel at an global summit in London.
"It delivers on the vote that people gave on Brexit, it delivers the fact that we will have an end to free movement, we will have an end to the jurisdiction of the European Court of Justice in the United Kingdom, we won't be sending vast contributions to the EU every year, we'll be out of the Common Agricultural Policy, out of the Common Fisheries Policy", she said.
Asked on BBC radio about Trump's comments ahead of his trip to Britain, he said: "There's always turmoil in every country but no, no, I think the U.K.is proceeding as it always does".
But senior pro-Brexit Cabinet ministers said they supported May and would not resign.
Mr Rees-Mogg has previously hit out at peers in the Lords for tabling amendments to Brexit legislation, accusing them of "playing with fire".
Asked about the balloon, Johnson said: "It's really kind of irrelevant to what we're trying to do and it certainly doesn't express the point of view of the British people I've met". Farage, who quit as leader following the 2016 European Union referendum result, called on Conservative MPs to hold a vote of no confidence over the prime minister.
"In that respect we are truly headed for the status of colony - and many will struggle to see the economic or political advantages of that particular arrangement", he said.
May's proposals for a future European Union relationship after Britain departs from the bloc next March had taken two years of internal wrangling to agree.
"We all know we want to avoid a no deal, cliff edge, hard Brexit, everyone seems to want to avoid that and if we come to an agreement that works for Ireland, UK and the entire European Union, then I think both sides will have to move a little".
The British government is due to publish a detailed version of its plans on Thursday.
German Chancellor Angela Merkel, speaking at the Western Balkans summit in London on Tuesday, said it was a "good thing" that there was a British proposal on the table.
The trans-Atlantic relationship has had some awkward moments since Trump's election.
Now the E.U.is waiting for May's position on whether she wants to follow Brexit hardliners' path of exit or try to maintain a relationship with the European Union despite leaving the organization.
Theresa May only has a majority in Parliament with the support of the 10 MPs from Northern Ireland's Democratic Unionist Party, so any split raises questions about whether her plan could survive a Commons vote.
The departure of Boris Johnson is unlikely to fix the strained ties all at once, since they've sunk way too low over the past few months. Gauke said. "The challenge is all very well for people to say I wouldn't do this".