Paul Thwaite, the head of the bank's commercial and institutional business, will helm NatWest for an initial period of 12 months, the company said in a statement.
NatWest has faced intense political and media scrutiny over a decision by its private bank Coutts to close Farage's accounts. An internal review obtained by the politician-turned-TV show host showed Coutts' wealth reputational risk committee had said his values did not align with the bank's own.
Some major NatWest investors are also applying pressure on the bank's chair Howard Davies to step down for mishandling the fallout, including saying Rose had the board's "full confidence" hours before she stepped down, the Financial Times reported.
In a post on the X social media platform on Wednesday, Farage called for further heads to roll in the wake of Rose's resignation.
"Others must follow," he said. I hope that this serves as a warning to the banking industry," he said, pledging to speak up for other people who had their accounts shut down.
NatWest shares were down 4% at 1358 GMT, the biggest faller in the FTSE 100 index.
A government official told Reuters ministers had been unhappy with the way Rose behaved and she did "the right thing" by resigning. NatWest remains 39% owned by taxpayers following its bailout in the 2008-9 financial crisis.
"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," said financial services minister Andrew Griffith on the X platform.
Keir Starmer, leader of the opposition Labour party, told a BBC radio phone-in that Rose was right to resign and said no-one should be refused banking services over their political views.
Data regulator the Information Commissioner's Office also said on Wednesday it had written to banks to remind them of their responsibilities to the public.
"Nigel Farage's experience shows why data protection rights remain so important," said Information Commissioner John Edwards, who added the watchdog would give NatWest a chance to respond to Farage's complaint before getting involved.
PERSISTENT SPECULATION
Rose, who had a more than 30-year career at the state-backed lender and was made a Dame for services to the financial sector, had faced days of speculation over whether she had discussed the account after the BBC ran a story on the topic.
On Tuesday, she acknowledged in a statement that she had discussed Farage's "relationship with the bank" with BBC Business Editor Simon Jack.
At the time, NatWest Chairman Davies said Rose retained the "full confidence" of the bank's board and there would be an independent review into the matter. But following an emergency board meeting late on Tuesday, the bank said she would step down.
"It is a sad moment. She has dedicated all her working life so far to NatWest and will leave many colleagues who respect and admire her," Davies said in a statement.
On Tuesday, Farage said on his eponymous TV show that Rose was "unfit" to run a bank. He also lambasted Davies and Coutts CEO Peter Flavel.
The BBC on Monday apologised to Farage, after originally reporting that his accounts were closed after he fell below the financial threshold required to be a Coutts customer.
The review later obtained and published by Farage revealed his political views were also a factor in the decision, alongside commercial considerations. Jack later described the original story as "incomplete and inaccurate".
In her Tuesday statement, Rose said she had not revealed any personal financial information about Farage in her discussions with Jack and had answered a general question about eligibility criteria required to bank with Coutts and NatWest.
She realised that her comments had left Jack with the impression that the decision to close Farage's accounts was solely a commercial one, Rose said in the statement.
Rose also said she was not part of the decision-making process to "exit" Farage's accounts and said this was a decision made by Coutts.
Britain's Financial Conduct Authority (FCA) has urged NatWest's board to review the matter.
"It is vital that the review is well resourced and those conducting it have access to all the necessary information and people in order to investigate what happened swiftly and fully," said Sheldon Mills, the FCA's executive director of consumers and competition.
Britain's financial services minister Andrew Griffith is set to meet lenders later on Wednesday to discuss concerns that banks have closed customer accounts over their political views. This comes ahead of reforms requiring banks to explain and delay these decisions.
($1 = 0.7765 pounds)
