Billy Richardson was one of the most highly regarded defenders in the League of Ireland at the time of the 1948 London Olympic Games, his performance captaining the Bohemians XI against Manchester United in May 1948 - described as 'a first-class exhibition at left back, his best of the year' - considered instrumental in his Olympic selection. He played for the Irish football team in their 3-1 defeat to the Netherlands at Fratton Park, Portsmouth on 26 July 1948. The match eliminated Ireland before the Opening Ceremony.
Billy Richardson was the son of Matthew Richardson, a former chairman of the St. James' Gate club. He played junior for Mountpleasant and made his first appearance for Bohemians in the League of Ireland in 1942-43. Richardson was eventually selected to play on the 1948 Irish Olympic football team alongside Bohemians teammates Paddy Kavanagh, Donald Lawler, Emmett McLaughlin, Bobby Smith and Brendan O'Kelly; the team would go on to lose in its only match to the Netherlands by a score of 3-1. He later remained with Bohemians until the 1949-50 season, appearing in a total of 159 matches with them in 7 seasons.