Anderson was born in Northern Ireland and moved to England in 1972 with ambitions of becoming a racing driver. He found work instead as a mechanic at Motor Racing Stables at Brands Hatch, then joined Brabham where he helped build their Formula 3 cars and progressed to become chief mechanic of the Formula One team. In 1975, together with mechanic Bob Simpson who worked at Tyrrell Racing, Anderson built a Formula 3 car called the Anson SA1, based on the Brabham BT38, which he raced in Formula Libre. Anderson left Brabham at the end of 1976 to design the Anson SA2 for 1977, but a last-minute withdrawal of sponsorship ended the project mid-season.
He rejoined Formula One as chief mechanic at McLaren, serving two years before moving to the Ensign team. Anderson and Simpson later relaunched the Anson project with competitive Formula 3 and SuperVee machinery. In 1985 he moved to the United States to become chief engineer of the Galles Indycar team, where he first worked with driver Roberto Moreno. The partnership proved fruitful when Anderson became technical director of Bromley Motorsport: Moreno won the 1988 International Formula 3000 championship using a Reynard 88D chassis. Adrian Reynard then hired Anderson to design the 1989 and 1990 Reynard F3000 cars.
In 1990, Eddie Jordan asked Anderson to design the first Jordan Grand Prix Formula One car. The Jordan Ford 191 made its debut in the 1991 season and proved unexpectedly competitive, scoring points finishes on seven occasions and finishing fifth in the Constructors' Championship. Anderson was appointed technical director from 1992 onwards. His Jordan Hart 194, co-designed with Andrew Green, brought the team its first podium at the 1994 Pacific Grand Prix with Rubens Barrichello. Anderson remained at Jordan for the rest of the decade despite reported interest from McLaren and Ferrari, until he was replaced by Mike Gascoyne midway through the 1998 season.
Anderson joined Stewart Grand Prix and designed the Stewart Ford SF3 for the 1999 season. It proved the team's most successful year: four podium finishes, including a victory at the 1999 European Grand Prix, and fourth place in the Constructors' Championship. Anderson later described team owner Jackie Stewart as the best person he had ever worked for. When Ford acquired Stewart and renamed the operation Jaguar Racing for 2000, Anderson stayed to design the Jaguar Cosworth R1, which scored two points finishes and finished ninth in the Constructors' standings. He was then replaced at Jaguar and spent 2001 working for Reynard in CART.
Anderson rejoined Jordan in 2002, co-designing the Jordan Honda EJ12 with Eghbal Hamidy. The team managed five points finishes. For 2003, with Honda engine support gone, Anderson's EJ13 ran Cosworth power badged as Ford and ended the season ninth — but it won the 2003 Brazilian Grand Prix, delivering Jordan their sole Formula One victory.
After leaving Jordan in 2003, Anderson transitioned to broadcasting. He worked for RTÉ from 2003 to 2005, Setanta Ireland from 2006 to 2009, and ESPN Star Sports as a technical analyst. In January 2012, the BBC recruited Anderson as technical analyst and pitlane reporter for its Formula One coverage. He later contributed technical analysis to Autosport magazine and joined the Formula One Management television production team in 2014. He was awarded an honorary doctorate of science by the University of Ulster in 2014. Anderson subsequently hosted a technical podcast for The Race alongside journalist Edd Straw and has written a technical analysis column for The Telegraph since 2022.