Adds Bailey comments in paragraphs 5-6, 11-13

By William Schomberg and Phoebe Seers

LONDON, July 14 (Reuters) - Bank of England Governor Andrew Bailey said on Tuesday he was concerned about the resumption of hostilities between the United States and Iran in recent days but so far there was no big impact on Britain's inflation outlook.

Bailey said last month, when he voted with the 7-2 majority on the BoE's Monetary Policy Committee to hold interest rates, that he thought that risks from the war in Iran were around the lower end of the central bank's three scenarios.

"But I would have added the very strong caveat to that it seems to me that the situation remained unstable, and the ceasefire was fragile," Bailey told lawmakers on the Treasury Select Committee.

'THE BIG ISSUE IS GROWTH'

"I think where I am now for the moment is that obviously that, as I say, that instability has come to pass," he said. "It underlines that this is going to be an unstable process for the foreseeable future."

Bailey said economic data pointed to limited impact on UK inflation. "We are seeing continued fairly soft evidence on the pass-through into UK prices," he said.

British government bond yields rose earlier on Tuesday to their highest level since May and investors ramped up their bets on interest rate hikes by the BoE as U.S.-Iran tensions pushed up oil prices and after a Federal Reserve official hinted at an increase in borrowing costs soon.

Asked whether the next change in Britain's prime minister would affect financial stability, Bailey declined to comment on politics but said the UK outlook was backed by the government's fiscal framework as well as monetary policy set by the BoE.

Former Manchester mayor Andy Burnham, who is on course to replace Keir Starmer as prime minister next week, has pledged to stick to the government's fiscal rules, but some investors are worried that he might want to increase public spending.

Bailey also stressed the importance of speeding up Britain's economic growth.

"I think the big issue is growth in the economy. So I think we can do all we can on financial stability," Bailey said.

Bailey used much of the hearing with lawmakers to reinforce the BoE's position in areas facing heightened regulatory scrutiny, including bank capital, leverage and artificial intelligence.

Pushed by lawmakers on industry calls to ease capital requirements to support economic growth, Bailey said: "memories of the financial crisis are now disappearing in the rear-view mirror ... we will not have a strong economy without financial stability".

He also described AI as a key risk for the banking sector, adding that the BoE was "very actively engaged" with banks on efforts to strengthen their cyber-defences.