Dibley was born in Hong Kong to Engineer Rear-Admiral Albert Kingsley Dibley and Penelope Dibley, née Frend, a descendant of Admiral Sir Hugh Palliser. The family returned to England in 1939 just before the Second World War. Dibley attended Pinewood school and Marlborough College, and showed an early interest in aviation sparked by proximity to the Royal Naval Air Station Lee-on-Solent, where his father's final naval posting was to the nearby Devonport dockyard.
He joined the Combined Cadet Force and transferred to the RAF section to pursue pilot training. After completing 30 hours in a De Havilland Tiger Moth, he received his private pilot's licence shortly after turning seventeen. He completed two years of National Service aboard HMS Ark Royal after deep defence cuts in 1957 ended Volunteer Reserve flight training.
On leaving the Navy, Dibley trained with Air Service Training at Hamble, obtained a Commercial Pilots Licence in September 1958, and was hired by British Overseas Airways Corporation (BOAC). He flew the Douglas DC-7C, Bristol Britannia, and De Havilland Comet 4 in his early career, then transitioned to the Boeing 707. In the mid-1960s Dibley joined the 707 Check Navigation Office, conducting annual competency checks on navigators.
When BOAC received its first Boeing 747 in 1970, Dibley became a flight simulator instructor and Type Rating Examiner. By early 1975, at age thirty-seven, he became the first of British Airways' First Officers to be given direct command of the 747. He also trained crews on the Lockheed L-1011 TriStar and was appointed Flight Technical Manager in 1977. In 1978 he chaired a newly formed Fuel Policy and Fuel Conservation Working Group, producing research on high-altitude descents and noise and fuel reduction that earned him the Guild of Air Pilots Brackley Memorial Trophy for 1979/80.
In 1984, while piloting a 747 on approach to Perth, Australia, a starboard engine exploded at 49 feet above the tarmac. Dibley diverted over water to dump fuel and landed the aircraft safely. He subsequently served as Director of Operations and Chief Pilot for Sheikh Zayed's private fleet in Abu Dhabi. After retiring from British Airways in 1992, Dibley held further roles as Chief Pilot at Air Hong Kong, Director Operations at Air Mauritius, and Flight Instructor at Airbus Training in Toulouse until 2002. He is a Fellow of the Royal Aeronautical Society and an elected member of the French Académie de l'air et de l'espace.
Dibley began racing in 1959 in an AC Aceca-Bristol coupé at a Silverstone meeting organized by the Nottingham Sports Car Club. In 1961 he moved to Formula Junior in a Lola Mk3 fielded by Écurie Light Blue, and the 1961 Lewis-Evans Trophy marked the first time a Lola appeared in a Formula 1 race. In 1962 he drove a Lola Mk5 in the Nassau Speed Week Formula Junior Championship, finishing second.
In 1964 Dibley switched to sports cars with a Brabham BT8, posting five wins and four lap records under the Stirling Moss Auto Racing Team (SMART) banner, and was awarded the inaugural "Goodwood Ton." From 1965 he drove Lola T70s in various specifications through 1969, and from 1967 to 1969 competed in the British Saloon Car Championship in a Chevrolet Camaro Z-28.
At the 1966 Can-Am season opener at Circuit Mont-Tremblant, during practice his Lola T70 backflipped, cleared a fence, and landed in the spectators' area. Dibley escaped unhurt; the manoeuvre became sufficiently notorious that any driver who subsequently repeated it was said to have "done a Dibley."
In 1968, Dibley partnered with Dick Thompson in the Howmet TX turbine-powered car at the BOAC 500 at Brands Hatch, and made his sole Le Mans appearance co-driving another Howmet with Bob Tullius, retiring with a wheel bearing failure. His final sportscar start came at the 1974 1000 Kilometres of Brands Hatch, where he finished third in a Mirage GR7.
In 1966 Dibley partnered with tool-maker Len Wimhurst — who had previously worked at Lola and Brabham — to design and build their own racing car. Wimhurst built the first chassis in his Catford backyard by December 1966. After early success, American dealer Bob Winkelmann became involved and placed orders for US sale.
Palliser Racing Design Limited was formally incorporated in October 1968, taking Dibley's second given name for the company. Production moved to North Street, Clapham. The cars followed a consistent naming convention: WDB (Formula B), WDF (Formula Ford), WDA (Formula A), WDV (Super Vee), and WD3 (Formula 3). Palliser also supplied components including cast wheels, magnesium uprights, and steering racks to other constructors including March Engineering. Over approximately four years the company built around 150 cars. Palliser cars won four championships, including the first Formula Atlantic Championship in England with Vern Schuppan driving. The company was sold in early 1972 and a liquidator appointed in March of that year.