He began a long career in motorsport began in 1961, driving a Mini in British club saloon racing. He then upgraded to a Lotus Elan and entered GT racing, scoring some excellent results, and then having a difficult time in Formula Three, where his natural speed was blighted by mechanical failures.
Nevertheless, for 1967 he was drafted into the Team Lotus Formula Two team, which also saw him making his Grand Prix debut in the F2 class at the German Grand Prix, where he came 5th overall and won the F2 class. In 1968 he was called up by Colin Chapman to take over the works F1 seat for Team Lotus after the tragic death of Jim Clark. The season would turn out to be difficult, with Oliver struggling for finishes. He led the British Grand Prix until an engine failure, and would only finish twice, his best result being 3rd place at the season-closing Mexican Grand Prix.
With Jochen Rindt signing for Lotus for 1969, Oliver was out, switching to BRM. He was to suffer disappointing two years at the Bourne team, which would effectively kill off his Grand Prix career. In two years he would muster just four finishes, with his only points scores being 6th place in the 1969 Mexican Grand Prix, and 5th in the 1970 Austrian Grand Prix. The vast majority of his other races saw the BRM break down.
His best results in these seasons would come from endurance racing, in John Wyer's Gulf Ford GT40, winning the Sebring 12 Hours and Le Mans 24 Hours events with Jacky Ickx in 1969, and the Daytona 24 Hours and the 1000km of Monza in 1970 with Pedro Rodriguez.
In 1970, he would debut in CanAm, initially for Autocast, and then for Don Nicholls' Shadow team. 1971 saw him out of a full-time Formula One drive, though he had three drives in a third McLaren. 1972 saw him concentrate mainly on CanAm with Shadow, though he would take a one-off drive for BRM at the British Grand Prix, where he retired.
For 1973, Shadow entered F1, and Oliver was nominated as team leader. The DN4 proved a difficult chassis, and once again his season was blighted by mechanical errors. However, in the Canadian Grand Prix he ran well, and many believe he actually won the race, but the lap charts were thrown into confusion by a rain shower meaning multiple pit-stops, and a staggeringly inept deployment of a pace car by the organisers. As it was, Oliver was classified 3rd, his only points finish of the year.
1974 saw Oliver concentrate on CanAm, taking the series title for Shadow. He was becoming more involved in the management side of Shadow, but would compete in Formula 5000 for the team for three seasons, and even briefly returned to F1, finishing 5th in the 1977 Race of Champions, and taking 9th in the Swedish Grand Prix.
At the end of the year he left Shadow along with financer Franco Ambrosio, designers Tony Southgate and Alan Rees, and engineer Dave Wass to form the Arrows Grand Prix team, also taking the young Riccardo Patrese to the new team.
Arrows would become famous for having the longest losing streak in Formula One history, 382 races with no wins. However, the team would always have well-presented cars which would usually be competitive, if not front-runners, and would often give breaks to talented drivers - besides Patrese, Thierry Boutsen, Gerhard Berger, Marc Surer and Martin Donnelly would all drive for the team early in their career.
Oliver sold much of his stake to the Japanese Footwork Corporation in 1990, remaining as director, but the team failed to move forward and the company pulled out at the end of 1993 due to financial trouble. Oliver had his team back, but money was tight, and in 1996 he again sold most of his shares to Tom Walkinshaw's TWR group. Oliver remained on the board until 1999, when he sold his remaining shares for a sizeable amount of money.