Attwood began his working life as an apprentice at Jaguar before taking up motor racing in 1960 with a Triumph TR3. In 1961 he joined the Midlands Racing Partnership (MRP) for club-level Formula Junior events, remaining with the team through 1962. When MRP moved into international Formula Junior competition in 1963, Attwood made an immediate impression by winning the Monaco Grand Prix Formula Junior support race in a Lola Mk5a. This result, along with his wider performances that year, earned him the inaugural Grovewood Award, voted by a Guild of Motoring Writers panel.
For 1964, MRP stepped up to Formula 2. Attwood won in Vienna and collected second places at Pau, the Eifelrennen, and the Albi Grand Prix — results achieved in a category where reigning Formula One World Champion Jim Clark regularly appeared. At Pau, Attwood finished second only to Clark's works Lotus.
These Formula 2 performances prompted BRM proprietor Alfred Owen to offer Attwood a works Formula One drive. His first outing, in a non-championship race at Goodwood, saw him finish fourth in the BRM P57. His second appearance came at the 1964 British Grand Prix, where he drove BRM's experimental four-wheel-drive P67 — primarily a test bed — which he qualified last but BRM withdrew before the start.
Tim Parnell signed Attwood to his Reg Parnell Racing privateer operation for 1965, racing a Lotus 25 with a BRM engine. The chassis was past its best by then and results were modest, limited to a pair of sixth-place finishes.
In the Tasman Series for 1966, Attwood showed strong form including a win at Levin, but he sat out most of the 1966 and 1967 Formula One seasons. His only F1 appearance came as a substitute for Pedro Rodríguez at the 1967 Canadian Grand Prix, finishing tenth in a Cooper-Maserati.
After Mike Spence died during Indianapolis 500 practice in 1968, Attwood rejoined BRM as his replacement. His most spectacular moment of this second stint came immediately: at the 1968 Monaco Grand Prix, he set the fastest lap on his way to second place behind Graham Hill's Lotus. Results faded later in the season and he was replaced by Bobby Unser with four races remaining. His final Formula One start came when Colin Chapman brought him in as a substitute for the injured Jochen Rindt at Monaco, driving the Lotus 49B to fourth place. Over his Formula One career he contested 17 World Championship Grands Prix, achieved one podium, and scored 11 championship points.
Attwood was involved in iconic machinery throughout his sports car career. As early as 1964 he was among the first drivers to test the Ford GT prototype that would evolve into the GT40, sharing one at Le Mans that year with Jo Schlesser before retiring with a fire. His first major international sports car win came at the 1964 Rand 9 Hours in South Africa, driving David Piper's Ferrari P2.
He maintained a long partnership with privateer David Piper over the following years, racing green Ferraris including the 250LM and 330P3/4. Highlights included third at the Spa 1000 km and second in the 500 km Zeltweg in 1967. He also drove machinery as diverse as the Porsche 906, Alfa Romeo T33, and the Ford P68 — the notoriously unreliable GT40 successor.
For the 1969 World Sportscar Championship, Attwood signed with the Porsche factory team. Regularly paired with Vic Elford, the season's highlights were a pair of second places in the BOAC 500 at Brands Hatch and the Watkins Glen 6 Hours, both driving the Porsche 908. He was also involved in the early development of the Porsche 917: the Elford/Attwood car completed 327 laps at the 1969 Le Mans before a gearbox failure ended their race two hours from the finish, despite leading for a substantial portion.
Attwood's defining achievement came at the 1970 24 Hours of Le Mans, where he and Hans Herrmann drove the Porsche 917K to victory. The win was Porsche's first at Le Mans and launched the manufacturer's unmatched run at the circuit. Herrmann and Attwood also took second at the 1970 Nürburgring 1000 km in a Porsche 908. The following year, paired with Herbie Müller in a Porsche 917 for the John Wyer privateer team, Attwood finished second again at Le Mans. He also won the 1000 km Zeltweg that season with Pedro Rodríguez, then retired from front-line motorsport at the end of 1971.
He made a brief return in 1984 as part of the Aston Martin Nimrod Le Mans project, but after a serious accident during the 1984 24 Hours of Le Mans — while John Sheldon was at the wheel — Attwood withdrew from front-line racing for good. He remained active in historic motorsport for many years, regularly appearing at Goodwood events with his personal Porsche 917, a car once used by Steve McQueen during filming of Le Mans. Painted to represent his 1970 Le Mans-winning livery, Attwood described it as "my pension" — a description borne out when he sold the car at auction in 2000 for £1 million.