Theresa May has faced taunts that her Chequers deal is “dead in the water” after caving in to a series of changes to customs rules demanded by pro-hard Brexit Tories.

Plans for the UK to collect duties for the EU – which lie at the heart of the prime minister’s hopes for a deal with Brussels – will only go forward if the EU in turn agrees to collect them for the UK.

Theresa May Photographer: Andrew Matthews/PA Images via Getty Images

There appears to be no prospect of the EU bowing to such a request, apparently throwing the hard-fought Chequers proposals up in the air after just 10 days.

In the Commons, Ms May was accused of “dancing to the tune of the European Research Group” – the 60-80 strong organisation of Brexiteer MPs led by Jacob Rees-Mogg.

“By capitulating to their proposals on the Customs and [the] Trade Bill she is accepting that the Chequers deal is now dead in the water,” said Labour MP Stephen Kinnock.

Ms May insisted he was “absolutely wrong”, telling MPs: “I would not have gone through all the work that I did to ensure that we reached that agreement only to see it changed in some way through these bills. They do not change that Chequers agreement.”

Nevertheless, the Brexit white paper – published only four days ago – appeared to rule out a requirement for the EU to agree reciprocal arrangements.

It said the two sides would have to “agree a mechanism for the remittance of relevant tariff revenue”, but added: “The UK is not proposing that the EU applies the UK’s tariffs and trade policy at its border for goods intended for the UK.”

Layla Moran, a Liberal Democrat MP and supporter of the anti-Brexit Best for Britain group, said: “First Trump humiliated the prime minister and this week it’s his British fan club – Theresa May has caved in yet again.”

The controversy blew up as the government refused to deny rumours that the summer recess could be brought forward – to head off any backbench plot to bring down the prime minister.

A motion will be brought forward on Tuesday to end the parliamentary term on Thursday, Commons sources have said, thereby ditching plans to sit on Monday and Tuesday next week.

Ms May’s Tory opponents are collecting signatures to force a vote of no confidence in her leadership of the Conservative party, with 48 required.

If MPs are sent on their summer holidays this week, it will probably be impossible to stage that contest – giving the prime minister respite until the autumn at least.

No 10 confirmed the government was accepting all four ERG amendments to the Taxation Bill, formerly known as the Customs Bill – just hours after Greg Clark, the business secretary, insisted it would not.

One would enshrine in law that there can be no customs border down the Irish Sea – proposed by the EU as the “backstop” to avoid a hard Irish land border, without which there will be no withdrawal agreement.

Although Ms May has ruled out a border in the sea, she has been inching towards accepting the backstop on the basis that it would never be needed.

Another amendment would require the UK to have a separate VAT regime from the EU, which would kill off any prospect of Britain remaining in a customs union.

A No 10 spokesman defended accepting the amendments – which were debated at last week’s Independent subscribers’ event – saying: “We believe they are consistent with the white paper we published last week.”

But the cave-in triggered an extraordinary breakdown of Tory discipline in the Commons, as pro and anti-EU Tories traded insults.

When Anna Soubry mentioned Margaret Thatcher, Brexiteer Edward Leigh said: “I knew Margaret Thatcher. I worked for Margaret Thatcher. My honourable friend ain’t no Margaret Thatcher.”

Ms Soubry condemned the ERG’s “wrecking amendments” and accused the prime minister of being “frightened of somewhere in the region of 40” hard Brexit supporters.

“Who is in charge? Is it the prime minister or is it the honourable member for North East Somerset [Mr Rees-Mogg] – I know where my money is sitting at the moment,” she added.

“This government is in grave danger of not just losing the plot, but losing a considerable amount of support from the people of this country.”

Peter Dowd MP, a Labour treasury spokesman, said: “It took two years for the prime minister to reach her Chequers deal, but only two days for it to fall apart.

“The white paper lies in tatters, the government is in free fall and Theresa May has no authority left.”