Cope played four times for England Schoolboys in 1950. He joined Manchester United as an amateur in the same year, turned professional in 1951, and made his debut for the club on 29 September 1956 against Arsenal at Highbury. However, he only gained a regular first-team place after the Munich air disaster. He lost his first-team place to Bill Foulkes in October 1960, and left United for Luton Town in August 1961 on a £10,000 transfer. He left the professional scene two years later, although he continued to play at non-league level for Northwich Victoria and also served the Cheshire club as manager.