Thomson established Atom Bank with chief executive Mark Mullen in 2014, with the provider subsequently raising more than £250m of capital.
The bank has now lent more than £1bn to small businesses and homeowners.
Rosewell has been one of the lender’s non-executive directors.
Thomson said: “Since having the idea for Atom bank in early 2012 it has gone from being a big idea disguised as a small bank to a big idea and a big bank.
“I am immensely proud that Atom bank has now completed over £1bn of lending to households and small businesses.
“This is a testament to the efforts of every person at Atom bank, and to the support of all of our shareholders.
“I am an entrepreneur and champion of the customer first and foremost.
“I specialise in disrupting, creating and building and I hope this will be my contribution to the Atom bank story.
“I have loved doing this with Atom but, as the bank moves into a new phase of development, I feel the time is right for me to step down from the role of chairman.
“As founder I remain fully committed to Atom bank’s success and will remain a shareholder.”
Thomson helped launch Metro Bank in 2010.
He plans to spend the following year travelling, as well as marketing a book.
Rosewell said: “It is thanks to Anthony’s vigour and dedication that Atom bank is what it is today.
“Anthony is the consumer’s champion, leading from the front to bring true competition to the industry and to change banking for good.
“It is with regret but also great admiration that the board have accepted his resignation, and I and my colleagues fully understand his reasons for leaving having brought the business to maturity and achieved so much so quickly.
“Today there are 300 people working for Atom bank who will miss Anthony’s unique wit and wisdom, but customers and stakeholders should know that he leaves the bank in great heart and with an exciting future in prospect.”