Bradshaw was an assistant coach at the University of Alabama under Bear Bryant and was on the staff that won the 1961 national championship.

[1] At Kentucky Bradshaw inherited a program that had won a championship in 1950 under Bear Bryant and had been moderately well-regarded under Blanton Collier but posted a record of 25–41–4 (.386) in its previous seven seasons.

Bradshaw's 1965 team defeated #10 Georgia, 28–10, and appeared bound for a bowl game, being ranked in the AP top ten for four weeks in September and November of that season.

Bradshaw also helped recruit Nate Northington, who became the first scholarship African-American athlete to play in an SEC game when Kentucky faced Ole Miss on September 30, 1967.

Bradshaw assistants who went on to be head coaches included Dave Hart (Pittsburgh), Leeman Bennett (Atlanta Falcons, Tampa Bay Buccaneers), Bud Moore (Kansas), Homer Rice (Cincinnati, Rice, Cincinnati Bengals), Charley Pell (Jacksonville State, Clemson, Florida) and Chuck Knox (Los Angeles Rams, Buffalo Bills, Seattle Seahawks).