The transfer of ownership from Spanish owner Ferrovial is expected to conclude in the coming weeks when Milner will take up the reins again.
Prior to his appointment, Milner has been working with Buckthorn and OEP in support of its acquisition of Amey.
The deal is worth £400m for the Amey infrastructure services and consulting business, which employs 11,000 people and has a turnover of £1.4bn.
Milner stepped down from his earlier chief executive role at the end of 2019, after four years as CEO and more than a decade with the business.
At the time he was thanked by Ferrovial chief López Soria for his commitment to Amey and Ferrovial, navigating the business through ‘some very challenging years’.