In April, Hodgkinson was announced as the new technical director, but it was unclear when he would be able to join because he was firmly contracted to Mercedes' own power unit division.
And with tensions between Red Bull and Mercedes quite high throughout the 2021 title campaign, there seemed little chance of an agreement being put in place to fast track matters.
However, on Friday, both Mercedes and Red Bull announced that Hodgkinson would be able to start in May this year after they came to a mutual settlement.
A short joint statement said: "Mercedes F1 and Red Bull F1 today reached an agreement regarding the appointment of Ben Hodgkinson.
"Under the terms of that agreement, Ben, who joined Mercedes in August 2001, will be free to join Red Bull Powertrains from 24 May 2022."
Hodgkinson had been head of mechanical engineering at Mercedes' High Performance Powertrains since 2017, having worked at the German manufacturer's Brixworth headquarters for 20 years.
Speaking at the time of his signing, Red Bull team boss Christian Horner said: "We are delighted to welcome Ben to Red Bull Powertrains as technical director. He comes to this hugely exciting project as a proven race winner and as an innovator capable of leading a like-minded team of highly skilled engineers."
Hodgkinson was not the only Mercedes team member who Red Bull poached, with five other staff members being announced as moving across last year.
Mercedes' head of manufacturing Steve Blewett is to become Red Bull's power unit production director.
Omid Mostaghimi, who is Mercedes' F1 electronics team leader, will be Red Bull's head of powertrains, electronics and ERS.
Pip Clode, who is Mercedes' F1 power unit concept team leader, is to become Red Bull's head of mechanical design ERS.
Anton Mayo, a Mercedes engineering team leader, will be head of power unit design ICE, while Steve Brodie, Mercedes' F1 trackside and final inspection manager, will become Red Bull's Group Leader ICE Operations.