A goalkeeper, Swindin was born in Campsall, Doncaster, Yorkshire. He played as an amateur for various local clubs, including Rotherham United, before turning professional in 1934 with Bradford City. He played 26 times for Bradford City, before being signed by Arsenal in 1936 for £4,000. He made his debut against Brentford on 3 September 1936, and played nineteen games in his first season. To begin with, his time at Arsenal was characterised by nervous and erratic displays, and he was made to share the goalkeeping spot with Alex Wilson and Frank Boulton. However, he played seventeen games in 1937–38, the most of Arsenal's three keepers, and won a First Division medal.
The Second World War interrupted his career somewhat, but Swindin continued to play through the war for Arsenal, whilst acting as a PT instructor for the Army. In 1945, he made a guest appearance for Southampton.
By the time first-class football had resumed after the war, he became Arsenal's undisputed No. 1, and stayed there for the next six seasons. By now, he had put his erracticness behind him, and he was a commanding keeper who was especially known for his aerial ability and assured handling of crosses, as well as his strong physical resilience. He won his second League title in 1947–48. After the arrival of Ted Platt in 1950, Swindin was forced to share the goalkeeper's spot for 1949–50, but kept his place to play in the 1950 and 1952FA Cup finals; Arsenal won the former against Liverpool, but lost to Newcastle United in the latter.
By 1952–53, Swindin was beginning to show his age, and another talented keeper, the WelshmanJack Kelsey had taken his first-team place. Nevertheless, Swindin played 14 matches that season as Arsenal won the title again, giving him his third Championship winner's medal. Despite his excellent form for Arsenal, he was never capped by England at senior level, with Sir Walter Winterbottom preferring Frank Swift and Bert Williams between the sticks. In all, he played 297 first-class matches (not including wartime games) for the Gunners.
Swindin moved to Midland League side Peterborough United as player-manager in 1954, and took his team to several famous FA Cup runs (which included getting to the Fourth Round in 1956–57 and three consecutive Midland League titles between 1956 and 1958. Peterborough would go on to win the title twice more after Swindin left, enough to win election to the Football League in 1960.
In the meantime, Swindin had returned to Arsenal in 1958 as manager, and his side initially started strongly, finishing third in 1958–59. However, the team soon flagged and spent the next three seasons in mid-table. Despite signing players such as George Eastham and Tommy Docherty, Swindin was unable to bring any silverware to the club, whilst up the road the club's rivals Tottenham Hotspur won the Double in 1961.
After retiring from football, Swindin first owned a garage in Corby, before emigrating to Spain, where he spent several years following his retirement before returning to his homeland. In the later years of his life he suffered from Alzheimer's disease. He died at Kettering in October 2005, aged 90 years.