Kennedy will leave the company in January having resigned after two years with the group.
Culshaw previously held a similar role with Volker Stevin for more than three years and before that worked with Carillion for many years following the acquisition of Alfred McAlpine.
The new appointment sees three former Carillion bosses now holding key positions at the highways unit after a series of senior replacements this year.
Eight months ago Duncan Elliott, who previously headed up Strategic Highways with Carillion for nine years, became managing director of Galliford Try Highways.
He was also joined by David Bates as commercial director for Highways who also came from Carillion where he held a similar role.
Elliott said: “Our business has grown at a phenomenal pace following recent bidding successes such as the Highways England DIP framework.
“Nick’s appointment will support further growth and strong business performance.”