Born in East Ham, Essex, Bueb started racing seriously in a Formula Three 500cc Cooper in 1953, graduating to the Cooper works team in 1955 when he finished second in the British championship. He made occasional starts in Grands Prix in 1957 with a Connaught and a Maserati run by Gilby Engineering. The following year he raced Bernie Ecclestone's Connaught at Monaco, and drove a Formula Two Lotus at the German Grand Prix.