Wirth attended Sevenoaks School from 1977 to 1984 before studying mechanical engineering at University College London, where he graduated with a first-class honours degree. He became the youngest-ever Fellow of the Royal Institution of Mechanical Engineers, a distinction reflecting his early standing in the engineering community.
In August 1989, Wirth co-founded Simtek Research with Max Mosley and Donald Hughes. The firm provided design, research and development services to the motorsport industry, growing from a single-employee operation in Wirth's home to a company with its own wind tunnel facility on the Acres Industrial Estate in Banbury, Oxfordshire. Clients included the FIA, the Ligier Formula One constructor, and numerous Formula 3000 and IndyCar teams. In 1990 Simtek designed a Formula One car for BMW's then-planned works entry; the project was cancelled but the design was later updated and sold to Andrea Moda Formula for the 1992 season. When Mosley became FIA president in 1992, he sold his share to Wirth.
In October 1993, Wirth decided to enter Formula One as a constructor, founding Simtek Grand Prix. He served as founder, owner and technical director of the team from its inception through to its voluntary liquidation in June 1995. The team's two seasons were marked by very limited resources, a ninth-place finish as its best championship result, and the death of driver Roland Ratzenberger during qualifying at the 1994 San Marino Grand Prix — the first driver fatality at a Grand Prix weekend in 12 years.
In 2009, Wirth's company Wirth Design partnered with John Booth of Manor Motorsport to create a new entry for the 2010 Formula One season. Richard Branson's Virgin group became title sponsor and the team raced as Virgin Racing, with Wirth appointed technical director. The car he designed, the Virgin VR-01, became the first Formula One car developed entirely through computational fluid dynamics without any wind tunnel testing during its design or construction — a significant technological claim in a sport where wind tunnel time had long been a primary competitive tool.
Wirth also designed Virgin's second car, the MVR-02, for 2011. Its performance disappointed, failing to close the gap to the midfield relative to the VR-01's debut. In June 2011, Virgin announced it had parted company with Wirth and abandoned the exclusively CFD-based development policy.
Following his departure from Virgin Racing, Wirth continued to develop Wirth Research as an engineering consultancy applying advanced simulation and CFD methods across automotive and motorsport projects, building on the methodologies he had developed through the Virgin Racing programme.