Hours after she was assured she had the full faith and backing of her board of directors, the chief executive of one of Great Britain’s largest banks, NatWest, announced early Wednesday morning that she was stepping down following allegations that she improperly discussed the affairs of a client of one of the bank’s subsidiaries with a journalist from the BBC.

The bank announced the departure of Alison Rose in a 2 a.m. statement Wednesday, hours after she admitted to making a “serious error in judgment” when she discussed the affairs of conservative British pundit Nigel Farage with a reporter from the BBC. The broadcaster apologized to Mr. Farage for the reporting, which turned out to be inaccurate, on Monday.

In a statement, Ms. Rose — the first woman to run a major U.K. bank — said she remained “immensely proud of the progress the bank has made in supporting people, families and business across the U.K., and building the foundations for sustainable growth.”

Mr. Farage has alleged that an exclusive private bank at London, Coutts & Co., summarily “debanked” him in early July and closed his accounts at the NatWest subsidiary because of his political views. Ms. Rose told the BBC reporter that the accounts were closed because Mr. Farage had fallen below the bank’s threshold for account balances. Coutts, known as the “Queen’s bank” because the British royal family has been a client for generations, is the favored financial institution for many in the English aristocracy.

It later came to light, however, that Mr. Farage’s sensational allegations had some basis in truth despite Ms. Rose’s denials. Mr. Farage obtained a 40-page dossier on him maintained by Coutts which stated that maintaining its relationship with him poses a “reputational risk” to the bank. The dossier said Mr. Farage’s conservative values did not align with the bank’s “inclusive” standards.

“Particularly given the manner in which he states (and monetises) those views — deliberately using extreme, hateful, and emotive language (often with a dose of misinformation) — at best he is seen as xenophobic and pandering to racists, and at worst, he is seen as xenophobic and racist,” the dossier said.

“He is considered by many to be a disingenuous grifter and is regularly (almost constantly) the subject of adverse media.”

Conservative politicians rallied behind Mr. Farage in recent weeks, questioning whether banks — especially one partially owned by U.K. taxpayers, as NatWest has been since the financial crisis of 2008 — had any business screening their customers based on their political views. They said cases like Mr. Farage and the others that have come to light since he first made his allegations had a chilling effect on free speech.

The U.K.’s top banking regulator, Andrew Griffith, was meeting Wednesday with the chief executives of more than a dozen banks to discuss the issue of customer screening and treatment of what are called in European banking regulatory circles “politically exposed persons,” or customers with high public profiles that deserve especially intense scrutiny. In a statement, Mr. Griffith said Ms. Rose’s resignation was the right thing to do.

“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,” Mr. Griffith said on Twitter. “That was and is always unacceptable. I hope the whole financial sector learns from this incident. Its role is to serve customers well and fairly – not to tell them how or what to think.”