Crosbie is due to take up the role of chief executive in the first half of 2022, the society added. She is currently chief executive of TSB, a non-executive director of Perth-based energy giant SSE and a member of the Prudential Regulation Authority's practitioner panel.
The change at the top comes after the announcement in September that after nearly six years in the role, current Nationwide chief executive Joe Garner intended to stand down from the board.
The selection process was led by Kevin Parry, the society’s deputy chairman and chairman elect.
Parry said: “Following a thorough and rigorous selection process, [Crosbie] emerged as the outstanding candidate to lead Nationwide. She brings significant banking experience combined with deep operational and technological knowledge - core skills that are needed to run a modern building society.
“She is a strong advocate of mutuality and supports Nationwide's core purpose and the societal role it plays. I look forward to working with Debbie to continue to enhance the services we provide to our members.”
Crosbie said: “Nationwide's mutual status, combined with its trusted brand and market-leading customer service, make it a purposeful and unique force for good. I'm delighted to be asked to lead the society in its next phase of growth.”
Current chairman David Roberts added: “Debbie will join a society that is in excellent health, a testament to the outstanding contribution that Joe Garner and the team have made over the last six years.”
Garner is to continue in his role as chief executive until Crosbie’s arrival.
Last month, Nationwide saw its profits more than double thanks to higher lending margins on mortgages approved during the pandemic.
Pre-tax profits for the six months to the end of September topped £853 million compared with £361m a year earlier, results from the high street mutual revealed.
The group said it had been buoyed by its decision to keep lending at the start of the pandemic while rivals pulled back.
Nationwide is the second-biggest mortgage lender in the UK after Bank of Scotland owner Lloyds Banking Group, with gross mortgage lending during the period rising by £5.5 billion to £18.2bn, although its market share dipped from 12 per cent to 11.4 per cent. More than £5bn was lent to first-time buyers.
Robin Bulloch, chief customer officer, has been appointed interim chief executive at TSB, leading the bank “with immediate effect” and will work with Crosbie on an “orderly handover”.
Bulloch is said to have been at the heart of “transforming the TSB customer experience”.
TSB chairman Nick Prettejohn said: “We are grateful to Debbie for her outstanding contribution to TSB and wish her well for the future. She leaves TSB on a strong footing and I’m delighted that Robin Bulloch, who has significant retail banking experience at the highest level, has agreed to lead the business until such time as a permanent appointment is made.”