McCullough was performance director at Aston Martin, where former Red Bull designer Adrian Newey is principal, and will replace Max Verstappen's race engineer Gianpiero Lambiase, the current head of racing who is moving to McLaren by 2028 at the latest.
"We're delighted to welcome Tom to the team at the beginning of 2027 and look forward to the wealth of experience and expertise he will bring to the role," Red Bull said.
Lambiase's move was announced in April. McLaren have said the 45-year-old will have a supporting role to team principal Andrea Stella.