Coton started his professional career at Birmingham City in 1978, having signed from Mile Oak Rovers the previous year. He made his Football League debut as a 19-year-old, on 27 December 1980 in the First Division match against Sunderland. His first touch of the ball was to save a penalty awarded after 54 seconds. By the 1982–83 season he had established himself as Birmingham's first choice goalkeeper.
Birmingham were relegated from the First Division at the end of the next season, but Coton found his way back into the top flight with a transfer to Watford, for a sum of £300,000. He soon replaced Steve Sherwood in Watford's goal and at the end of his second full season at Vicarage Road he won the Hornets' Player of the Season and Display of the Season awards, the latter for a clean sheet against Liverpool. He remained with the club even after their relegation from the First Division in 1988. He went on to become Watford Player of the Season for an unprecedented third time in 1989–90.