Cookson was born in Manchester. He began his career as a wing half with South Salford Lad's Club and then played for Clayton and Manchester North End. He turned professional with Manchester City in August 1923, but was unable to break into the first team and was sent to Southport on loan for a trial period in 1924. He was transferred to Chesterfield in April 1925, and converted to a centre forward. He was the leading goalscorer in the Third Division North for the 1925–26 season with 44 goals, and scored 85 overall in 74 league appearances. In August 1927, he joined West Bromwich Albion for a £2,500 fee. He continued to score goals at his new club and was a member of the 1930–31 squad that won the FA Cup and promotion to the First Division. Later in 1931, Cookson was selected for the Football Association tour of Canada.