Theresa May has insisted she can achieve a “smooth and orderly Brexit”, at a press conference with Angela Merkel on a day when two Conservative party vice-chairs resigned in protest at her soft Brexit negotiating plan.

The prime minister sought to reassert her authority over her party as she appeared with the German chancellor, but she also had to fend off questions about whether Donald Trump was right to suggest the UK was in “turmoil”.

May looked uncomfortable as she addressed journalists at the press conference but she said the Brexit white paper expected on Thursday would keep “faith with the vote of the British people” and the UK would depart the EU in a “smooth and orderly” manner.

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She won a modest endorsement from Merkel, who said it was “a good thing we have proposals on the table” – referring to the plan agreed by the cabinet at Chequers last Friday, under which the UK would sign up to a “common rulebook” with the EU for food and goods as part of a wider Brexit deal.

Two cabinet ministers, Boris Johnson and David Davis, quit the government within the space of 24 hours at the start of the week. Davis said he feared May had conceded too much too early in the negotiating process, while Johnson said matching EU rules on food and goods left the UK heading “for the status of a colony”.

Johnson kept his counsel on Tuesday but rightwingers in the Tory party maintained pressure on the prime minister by coordinating the resignations of Ben Bradley and Maria Caulfield a few minutes before May and Merkel’s press conference in London was due to begin.

In his resignation letter, Bradley, the MP for Mansfield, said the Chequers proposals would leave the UK “tied to EU trade regulations” and he believed the outcome would be “the worst of all worlds” because it would “not deliver a Brexit in spirit”.

Caulfield, the MP for Lewes, wrote: “I cannot support the direction of travel in the Brexit negotiations, which in my view do not fully embrace the opportunities that Brexit can provide.”

She added: “The policy may assuage vested interests, but the voters will find out and their representatives will be found out. This policy will be bad for our country and bad for the party. The direct consequences of that will be prime minister Corbyn.”

Conservative party vice-chairs are not government ministers, although they are paid a salary of up to £10,000. There were eight vice-chairs before the resignations were announced, reporting to the party chair, Brandon Lewis, and the deputy chair, James Cleverly.

Hard Brexiters in the party had threatened May with a “drip, drip” of resignations that would undermine her premiership after Friday’s Chequers agreement. Five ministers, including Davis, Johnson and the influential Steve Baker, a junior Brexit minister, quit on Sunday night and Monday.

On Tuesday Michael Gove fended off repeated questions as to whether he would follow his fellow Brexiter Johnson out of the government. The environment secretary was asked by ITV as he left his house in the morning whether he would quit, and he replied: “Absolutely not.”

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At cabinet, May briefly thanked Johnson and Davis for their work, but there was no discussion of Johnson’s incendiary resignation letter of Monday in which he said the Chequers plan “sticks in the throat”. No 10 said there was further discussion of the EU divorce talks, led by the new Brexit secretary, Dominic Raab.

Raab also attended a scheduled meeting with five business organisations alongside the business secretary, Greg Clark, as ministers tried to move on from Johnson’s “fuck business” comments.

One of the organisations attending, the Federation of Small Business, said: “The Brexit secretary’s decision to go ahead with today’s planned meeting, despite only being in post for 24 hours, sends a good signal that he is keen to get on with the job in front of him.”

The EU’s chief Brexit negotiator, Michel Barnier, struck an emollient tone on a visit to New York, highlighting how far the talks had progressed. “After 12 months of negotiations we have agreed on 80% of the negotiations,” Barnier said, adding that he was determined to reach agreement on the remaining 20% by October or November.

Jamie Dimon, the chief executive of JP Morgan, said Brexit would hurt British jobs. In an interview with an Italian newspaper, he said: “I do think that, because of Brexit, some businesses across the financial and manufacturing sectors will be relocating from the UK to other parts of Europe, including Italy.”

A Labour source said: “With just weeks to go to negotiate Brexit, NHS waiting lists growing and pay packets being squeezed, the Conservative party continues to tear itself apart. Something has got to give.”

