Bradshaw is the son of the Delgany professional golfer Ned Bradshaw. He and his three brothers, Jimmy, Eddie and Hughie, all become professional golfers. He represents Ireland in the Triangular Professional Tournament at the Cawder Golf Club in Bishopbriggs, Glasgow, Scotland in October 1937 and the Llandudno International Golf Trophy match play tournament at the Maesdu Golf Club in Llandudno, Wales in September 1938. He wins the Irish PGA Championship ten times between 1941 and 1957, tied with Christy O’Connor Snr for most wins in that event. He is also the Irish Open champion in 1947 and 1949. He teams with Christy O’Connor to win the Canada Cup for Ireland in Mexico City, Mexico in 1958, finishing second in the individual section of the event despite suffering nosebleeds due to the altitude. He plays in the Ryder Cup in 1953, 1955 and 1957 and is twice Dunlop Masters champion, in 1953 and 1955.
Bradshaw loses the 1949 The Open Championship following a playoff against Bobby Locke at Royal St. George’s Golf Club, after an extraordinary incident in the second round when his drive at the 5th hole comes to rest against broken glass from a beer bottle on the fairway. Rather than taking a drop (to which he is probably entitled) he elects to play the ball as it lay, but is only able to move it slightly forward, dropping the shot. The setback results in his tying with Locke with an aggregate of 283, thereby equaling the championship record. However he loses the playoff to Locke. Arguably the incident with the bottle costs Bradshaw the tournament.
Bradshaw dies at the age of 77 on December 22, 1990.