Bank bosses have made a commitment to free speech, according to the government, in the wake of the Nigel Farage de-banking row that claimed the scalp of NatWest chief executive Dame Alison Rose.

On Wednesday afternoon the Information Commissioner’s Office announced it has written to banks to remind them of their “responsibility to the public”.

“Banks should not be holding inaccurate information, they should not be using information in a way that is unduly unexpected, and they should not be holding any more information than is necessary,” the Information Commissioner John Edwards said.

Dame Alison’s four-year tenure as chief executive ended in ignominy last night following her admission that she had discussed Mr Farage’s bank details with a BBC journalist, suggesting too that his account at the bank’s Coutts division had been closed only for commercial, rather than any political, reasons.

“Any suggestion that this trust has been betrayed will be concerning for a bank’s customers, and for regulators like myself,” Mr Edwards said.

Number 10 said Dame Alison had “done the right thing” by resigning and confirmed she was no longer a member of the prime minister’s business council. She has also left two roles she had with the department for energy after the secretary of state asked her to step down from both positions.

Treasury minister Andrew Griffith met 19 bank bosses for a summit on Wednesday to discuss concerns other figures, not just Mr Farage, were being denied access to banking due to their politics or perceived beliefs.

Mr Griffith said afterwards: “It’s not the job of banks to tell us what to think, or what political party we should support.

“The government’s been extremely clear on this, in a democracy that relies upon freedom of expression… that is not a legitimate thing for a bank to remove someone’s access to a bank account.”

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A readout of the meeting’s conclusions suggested the industry had agreed to work with government and regulators on the implementation of new rules aimed at strengthening protections on account terminations or access to accounts.

“Attendees from the sector acknowledged that recent events had impacted upon public trust for the whole sector and expressed their clear commitment to government policy on account closure and to act quickly to restore confidence,” the document said.

Mr Farage told Sky News “the whole board needs to go” at NatWest following the resignation of Dame Alison.

The former Brexit campaigner said Sir Howard Davies, chairman of the NatWest Group, had continued to endorse Dame Alison even after it emerged she was the person who had leaked to the BBC.

Sky’s City editor Mark Kleinman suggested it was unlikely Sir Howard would follow her out of the bank despite intense pressure on his own position, saying it could even be prolonged beyond his planned departure next year given the search for Dame Alison’s successor and the need for stability at the top of the bank.

“The first rule of banking is you have to obey client confidentiality. So they have made a complete and utter mess of this,” Mr Farage said, adding he had not decided whether he will seek compensation and the row over his account closure has “absorbed my life for many months”.

He said a subject access request from the NatWest Group revealed his account was “commercially viable” and its closure was a “political decision”.

The former UKIP leader also said he hadn’t been able to open another bank account and claimed he has been turned down by 10 banks.

Mr Farage also claimed he has been “approached by literally thousands of people all over this country that have been unfairly closed down by NatWest”.

NatWest’s shares were down by 4% following the news of Dame Alison’s resignation and were leading the fallers on the FTSE 100.

Mr Griffith earlier tweeted it is “right that the NatWest CEO has resigned”.

He added: “This would never have happened if NatWest had not taken it upon itself to withdraw a bank account due to someone’s lawful political views. That was and is always unacceptable.”

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NatWest chairman says resignation is a ‘sad moment’

Sir Howard said earlier the board and Dame Alison agreed by “mutual consent” that she would step down from her role.

He said it was a “sad moment” and that Dame Alison has “dedicated all her working life so far to NatWest”.

In a statement, Dame Alison said: “I remain immensely proud of the progress the bank has made in supporting people, families and business across the UK, and building the foundations for sustainable growth.

“My NatWest colleagues are central to that success, and so I would like to personally thank them for all that they have done.”

The resignation was expected in the wake of briefings by Downing Street that she had lost the confidence of the prime minister and chancellor

Their concerns were echoed by Mr Farage, who accused the management of Coutts bank – which is owned by NatWest – of a “serious breach” and called Dame Alison’s position “totally untenable”.

The story first came to light when the BBC inaccurately reported Mr Farage’s account was closed as he did not meet Coutts’s financial thresholds.

Documents obtained by Mr Farage subsequently showed his political beliefs and connections formed part of the rationale.

Mr Farage told Sky News he has written to Peter Flavell, head of NatWest’s Coutts unit, “three times” since his account was closed and had not even had the “courtesy of an acknowledgement”.

Dame Alison had said she believed it was public knowledge Mr Farage was a customer of private bank Coutts and had been offered a NatWest account, and so confirmed these details to BBC business editor Simon Jack.

She later called her actions a “serious error of judgement” but reiterated the bank saw the account closure as a commercial decision and she was not part of the decision-making process.

On Monday, the BBC apologised for the report, following earlier apologies from both Coutts and Dame Alison.

Paul Thwaite, the current chief executive of the company’s commercial and institutional business, was announced as an interim chief executive, for an initial period of 12 months, pending regulatory approval.

The board said a process to appoint a permanent successor will take place in due course.