Theresa May described a meeting of her new Cabinet as "productive" on Tuesday following the shock resignations of Boris Johnson and David Davis over the Prime Minister's Chequers Brexit plan.

Senior ministers including staunch Eurosceptic and Secretary of State for Environment, Michael Gove, rallied round Mrs May as she faced increasing pressure over Monday's resignations.

Jeremy Hunt, appointed the new Foreign Secretary as the Prime Minister carried out a hurried reshuffle of her top team, vowed he would be "four square" behind her in driving through her Brexit plan.

Among Tory Brexiteers there was deepening anger at the proposals agreed at Friday's meeting of the Cabinet at Chequers which they branded as "Brexit in name only".

However, it was unclear whether they had the numbers to force a leadership challenge.

Under party rules, 48 Tory MPs – 15% of the party’s 316-strong representation in the Commons – must write to the chairman of the backbench 1922 Committee, Sir Graham Brady, to trigger a no-confidence vote.

Sir Graham has consistently refused to say whether he had received any such letters.

After Mrs May addressed the 1922 Committee in Westminster on Monday about the Chequers plan, allies said that just six MPs expressed dissent in the course of the meeting.

Solicitor General Robert Buckland said there had been a realisation that "we all hang together or we all hang separately".

Nevertheless, many Brexiteers were furious after Downing Street said it would be be briefing Labour MPs on the Chequers plan.

Jacob Rees-Mogg, the leader of the influential pro-Brexit European Research Group of Tory MPs, said: "If the Government plans to get the Chequers deal through on the back of Labour Party votes that would be the most divisive thing you could do.

"And it would be a split coming from the top, not from the members of the Conservative Party across the country."

He made clear he had not submitted a letter of no confidence and expected Mrs May to remain at least until the official date of Brexit in March 2019.

However the Daily Mail reported Mr Rees-Mogg said Mr Johnson would make a "brilliant" prime minister.

Mr Johnson's resignation bombshell came less than 24 hours after Brexit Secretary David Davis announced he was quitting, saying he could not make the case convincingly for the Chequers proposals in the negotiations with Brussels.

Brexit minister Steve Baker and unpaid parliamentary aides Conor Burns and Chris Green also resigned.

In a scathing resignation letter, Mr Johnson – who led the official Leave campaign in the EU referendum – said the dream of the Brexiteers was "dying, suffocated by needless self-doubt" and that Mrs May’s plan would reduce the UK to the "status of a colony".

His replacement by Mr Hunt, a Remainer in the referendum campaign, is likely to further antagonise some Brexiteers.

It leaves the four great offices of state – Prime Minister, Chancellor, Foreign Secretary, and Home Secretary – in the hands of ministers who voted to stay in the EU, although in the case of Home Secretary Sajid Javid at least, with no great enthusiasm.

Boris Johnson leaves Carlton House Terrace in Westminster, London, after resigning as Foreign Secretary Credit: PA

In other moves, Culture Secretary Matt Hancock takes over from Mr Hunt at Health and Social Care. He in turn is succeeded by Attorney General Jeremy Wright, who is replaced by Geoffrey Cox.

Earlier Downing Street announced that Housing Minister Dominic Raab, a staunch Brexiteer, was taking over from Mr Davis as Brexit Secretary.

The turmoil comes at the start of a momentous week for Mrs May on the world stage with her attendance at the Nato summit in Brussels on Wednesday and Thursday followed by Donald Trump’s first visit to the UK as US president.

Mrs May will also host a summit of Balkan leaders on Tuesday.

Mr Hunt said it was a moment to show that Britain remained a "strong, confident voice in the world".

"My principal job at a time of massive importance for our country is to stand four square behind the Prime Minister so that we can get through an agreement with the European Union based on what was agreed by the Cabinet last week at Chequers," he said.

"This is a time when the world is looking at us as a country, wondering what type of country we are going to be in a post-Brexit world.

"What I want to say to them is Britain is going to be a dependable ally, a country that stands up for the values that matter to the people of this country, and will be a strong, confident voice in the world."