Senior figures from across industry say PM did not secure mandate in election to leave the single market

Senior business figures have heaped further pressure on Theresa May to change course for a softer Brexit in the wake of the election, amid fresh warnings of the impact of immigration controls and leaving the single market.

Stuart Rose, the Tory peer and chairman of online grocer Ocado, who backed Remain, said the election had been a “proxy re-referendum” against hard Brexit. Karan Bilimoria, the founder of Cobra beer and Remain backer, said the prime minister had “zero credibility” and that Britain could now rethink leaving. And Brian McBride, chairman of ASOS and Wiggle, has raised concerns about access to labour and customs checks.

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Business groups are pressing Downing Street for a change in tone, as well as a formal role in the Brexit process. They also want an end to May’s repeated mantra that “no deal is better than a bad deal”. Josh Hardie, the CBI’s deputy director-general, said: “It is a time for reflecting on how businesses are feeling. It is absolutely clear that for the new government, the priority has got to be putting the economy first.”

The intervention comes on the eve of Brexit talks that begin even before May has finalised a governing deal with the DUP or navigated a curtailed Queen’s speech through parliament. The chancellor, Philip Hammond, who had been in line for the sack before the election result, is also signalling he wants a softer Brexit that could include a more comprehensive transitional deal. In comments putting him at odds with the prime minister’s determination to go for a hard Brexit that includes leaving the single market, customs union and slashing immigration, he will today assert that the majority of Britons want a deal that protects jobs, economic growth and prosperity.

David Davis, the Brexit secretary, says on Sunday that he would head into Monday’s Brexit talks aiming to “secure a deal that works for all parts of the United Kingdom”, but signalled that the plan was still to leave the customs union and reduce immigration.

“We will soon introduce bills for new immigration and customs arrangements, and the great repeal bill will transpose all EU law into UK law,” he said. The bill, which could be hit by a series of parliamentary rebellions, will be introduced as part of the Queen’s speech on Wednesday.

The government also risks accusations of weakness by announcing that the parliamentary session will be extended to two years, effectively avoiding another Queen’s speech next year. Insiders insisted the move would give MPs “time to consider crucial legislation” on Brexit. They said it will still be a “substantial legislative programme” designed to tackle social injustice.

Meanwhile, British businesses have sounded the alarm over damaging labour shortages after Brexit, with thousands warning that they will be hurt by even a modest move away from the EU’s free movement rules.

Almost a quarter of the business community say that restricting entry to Britain to EU migrants with a job offer would have a negative impact on them, rising to 30% of companies in low-wage industries. The warning is contained in a comprehensive analysis of business views on migration by the Chartered Institute of Personnel and Development (CIPD) and the National Institute of Economic and Social Research to be published on Monday. The report contains alarming findings, revealed exclusively to the Observer, over the number of companies who believe they will be damaged by a move away from freedom of movement rules.

It includes a poll of more than 1,000 companies, weighted to represent Britain’s business community, which finds that 23% believe allowing in an unlimited amount of EU migrants with a job offer would have a negative impact on their organisation.

More than a third (35%) of businesses in low-wage industries said they recruited EU nationals because they cannot fill the positions with UK-born applicants, signalling that labour shortages could result from a clampdown.

There was even more disquiet about stricter schemes among the 91% of companies that expected to hire EU migrants. Some 28% warned that giving skilled EU workers five-year visas would hurt them. More (30%) said that one-year visas for unskilled workers would do damage.

A 10th of businesses warned that the numbers of EU nationals they were hiring was already falling in the wake of Brexit. Peter Cheese, chief executive of the CIPD, said: “Businesses are clearly sounding the alarm on the impact that constraints on immigration could have on the economy.”

The shifting political landscape has led some Remain-supporting business figures to speak out. Rose said: “I believe [the election] this was a proxy re-referendum. I think it was the public’s way to demonstrate that all was not all as they wanted. Is this an opportunity to have a rethink?

“I think [the single market and customs union] should still be on the table. Who is going to tell the man on the Clapham omnibus, and when will they tell him, that he is not going to be better off?”

Bilimoria said: “We have a prime minister who has zero credibility – with the country, with her party and in parliament. She was the driving force in pandering to the Brexiters in her party who were in the driving seat.”