The son of a successful motor trader, Attwood starting racing in a Triumph TR3 in 1960. In 1963 he switched to Formula Junior and sprang to international prominence by winning the 1963 Monaco Formula Junior race in a Lola entered by the Midland Racing Partnership, a group of enthusiastic racers from his part of the country who banded together to go racing. He had some success in 1964 in Formula 2 with a win at Aspern in Austria and also raced a Ford GT sportscar.In 1965 Attwood drove for Reg Parnell Racing in a Formula 1 Lotus-BRM in several Grands Prix and he finished sixth in both Italy and Mexico. In 1967 he appeared in Formula 1 just once at the wheel of a Cooper-Maserati in the Canadian Grand Prix and then joined the works BRM team in 1968, replacing Mike Spence, who had been killed at Indianapolis. The highlight of his season was a superb chase to second place and fastest lap at Monaco behind Graham Hill's winning Lotus. Dropped from the BRM team at the end of the year after a string of disappointing performances in other events, Attwood returned to Monaco in 1969 to finish fourth for Lotus as a stand-in for Jochen Rindt who had been injured in the Spanish Grand Prix. That was the end of Attwood's Formula 1 career and, after winning Le Mans in 1970 sharing a Porsche 917 with Hans Herrmann, he retired from the sport in 1971.