Swindin was born in Campsall in the West Riding of Yorkshire, and attended South Grove Central School in Rotherham. He played for England against The Rest as a final trial for a schoolboy international against Wales in 1929, but was not selected. He played as an amateur for various local clubs, including for Rotherham United's reserve team in the Midland League, before turning professional in 1934 with Bradford City. Swindin established himself in the first team towards the end of the 1933–34 season, but a serious knee injury sustained in the last match of that campaign caused ongoing problems that disrupted his second season and eventually required surgery to remove cartilage. He played 26 Second Division matches for Bradford City, before being signed by Arsenal in April 1936 for £4,000.