The Englishman began racing in 1959 and collected his first of four Manufacturers Championships in 1968, driving a Ford GT40 with Belgian Jacky Ickx for John Wyer Automotive Engineering.

Redman raced F1 for McLaren, Shadow, Cooper, Williams He participated in 15 World Championship Grands Prix, debuting on 1 January 1968.

He then had an accident at the Belgian Grand Prix at Spa-Francorchamps, crashing his Cooper-BRM when the suspension broke at the Les Coombes corner; he survived with a badly broken arm.

In 1973, Redman started racing in the USA full-time and won the SCCA/USAC Formula 5000 Championship three times in a row from 1974 to 1976 driving a Jim Hall/Carl Haas Lola T330/332C against considerable opposition, including Mario Andretti and Al Unser, Jody Scheckter, Jackie Oliver, Alan Jones, David Hobbs, Brett Lunger, Sam Posey, "Vern" Schuppan et al. Redman was employed at Haas's business office in Highland Park, Illinois, where he also lived for a time.

On the first day of practice for the first race of the new season, at Circuit Mont-Tremblant, Canada the new car took off at 160 mph went 40 feet in the air and turned over, landing upside down.