Born in Bushey, Bunce grew up in the local area, representing Hertfordshire as a schoolboy.
His first competitive appearance came on 22 October 1955, in a 2–0 English Football League Third Division South defeat at Southampton.
Nonetheless, during this period Bunce became the first Watford player to play for England at youth level since Fred Fayers in 1910.
[1] Bunce established himself as a frequent member of the first team squad in 1958–59, making 26 appearances in all competitions.
After missing the first three games of the following season, Bunce scored his first career hat-trick, in a 6–1 win over Brentford at Vicarage Road on 30 August 1960.