Born in Kidderminster in 1909, Wyer rose to prominence as team manager of Aston Martin's racing programme. He spent ten years with the team, and in 1959 — his tenth anniversary as manager — he guided Aston Martin to victory at the 24 Hours of Le Mans with Roy Salvadori and Carroll Shelby driving the DBR1. The team also won the 1000 km Nürburgring three years in succession during his tenure.
In 1963 Wyer left Aston Martin for Ford Advanced Vehicles, where he took charge of the Ford GT40 programme. The car's early years were troubled by chronic reliability problems and mechanical failures. Following disappointing results in the first two seasons, Ford transferred the competition programme to Holman Moody and Carroll Shelby, who went on to win the 1966 and 1967 24 Hours of Le Mans with the car.
When Ford closed Ford Advanced Vehicles after the 1966 season, Wyer and John Willment formed J.W. Automotive Engineering (JWA) to take over the Slough factory and continue building production GT40s on Ford's behalf. Despite the initials being those of John Willment, the team became universally associated with Wyer due to his visible presence at circuits. With backing from Gulf Oil, Wyer created the Mirage M-1, a Ford-powered prototype that won the 1967 1000 km Spa.
A rule change for 1968 restricted large-engine prototypes to 3,000 cc, but permitted sports cars of up to 5,000 cc if at least 50 examples had been built. This reprieve allowed Wyer to campaign the two-year-old GT40, which he had modified extensively. In a result that surprised the paddock — the 2,200 cc Porsche 907 had been considered the favourite — Wyer's team won the World Sportscar Championship for Ford in 1968, with the powerful 302-cubic-inch V8 GT40s dominating fast circuits and winning Le Mans in back-to-back years: in 1968 with Pedro Rodriguez and Lucien Bianchi, and in 1969 with Jacky Ickx and Jackie Oliver.
As the GT40 became obsolete under evolving regulations, Porsche took advantage of a loophole by building over two dozen 5,000 cc Porsche 917 prototypes, homologated as sports cars. Though the 917 was fast in 1969, it remained unreliable and had severe handling problems. After the season Wyer switched to the 917 and 908, and JWA became Porsche's primary factory partner. The team played a central role in developing the wedge-shaped Kurzheck tail of the 1970 917K, which transformed the car's stability and made it one of the most dominant machines in endurance racing history.
Driven by Jo Siffert, Brian Redman, Leo Kinnunen, Pedro Rodriguez, Richard Attwood, Herbert Mueller, and Derek Bell, the Gulf-Porsche 917s won seven of Porsche's nine victories in the 1970 season and five of eight in 1971. The team's best Le Mans result during this period was second place in 1971. A Gulf-Porsche 917K's fictionalised Le Mans victory in Steve McQueen's film helped make the Gulf colours among the most recognised in motorsport.
After Porsche withdrew from European-style sportscar racing when the rules changed again, Wyer adopted the new 3.0-litre regulations and developed a new series of Gulf-Mirage prototypes using the Cosworth DFV Formula One engine. After three years of attempts, Jacky Ickx and Derek Bell gave Wyer his final and perhaps most hard-earned Le Mans victory in 1975.
Wyer retired from competition the following year, selling JWA to Harley Cluxton's Grand Touring Cars operation. He died in Scottsdale, Arizona, on 8 April 1989, aged 79. His career record at Le Mans alone — victories in 1959, 1968, 1969, and 1975 — places him among the most successful team managers in the race's history.