The appointment comes as parent company National Australia Bank prepares to sell its UK-based operations to focus on its core business in Australia and New Zealand.
Mr Grace will join the board of both institutions, as well as the board of National Australia Group Europe, as non-executive director.
He was appointed chief executive of Aegon UK in 2011, having previously been chief operating officer and business development director of the insurer. Prior to this, he held senior roles with Barclays Insurance Services and Lloyds TSB.
It is the latest in a number of board changes in the last six months that included the appointment of Jim Pettigrew as chairman.
Mr Pettigrew said Mr Grace has a “first-class track record with outstanding business knowledge” that will benefit the banks as they move forward.
Earlier this week, Mr Thorburn announced his intention to leave his post as head of Yorkshire and Clydesdale banks.
NAB’s plans to accelerate their UK exit would likely require a five-year commitment to the role, something he saw as a “significant undertaking”, he said.
A search for Mr Thorburn’s replacement is already “well underway”, the banks said.