 Allcock was born in Leeds and began his football career with Bolton Wanderers, who approached him while he was captain of Yorkshire schoolboys.When interviewed in 2004 for the book 'Twelve Canary Greats', Allcock stated that his decision to sign for Bolton was made when his Father's job moved to Blackpool and Bolton permitted him to commute from there.He made his first team debut in October 1953 against Manchester City and scored twice in the first twenty minutes.