Crossley had taken up racing before the Second World War and resumed the activity in the immediate aftermath, competing in several events including the 1947 British Empire Trophy on the Isle of Man with a pre-war Alta. In 1949 he made a more serious commitment to the sport by ordering a new purpose-built Alta Grand Prix car from the Surrey manufacturer — chassis designated GP2 — which featured sleeker, more aerodynamic bodywork than earlier Alta examples.
In late 1949, Crossley used the new Alta to establish International Class F standing-start speed records at the Montlhéry autodrome outside Paris. He set three marks: 125.92 mph over 50 kilometres, 124.49 mph over 50 miles, and 124.17 mph over 100 kilometres. The records demonstrated the car's straightline speed on Montlhéry's banked oval.
He also entered the car competitively that year, including at the 1949 Belgian Grand Prix, where he finished seventh and last — six laps adrift of the winner — a respectable result for a privateer against the works opposition of the day.
When the FIA launched the Formula One World Championship in 1950, Crossley was among the privateer entrants who gave the inaugural season much of its numerical strength. At the opening round, the British Grand Prix held at Silverstone on 13 May 1950, he qualified seventeenth on the grid, three positions ahead of Joe Kelly in a comparable Alta despite Kelly running a higher-specification engine. Crossley's race ended on lap 43 when the transmission failed; he scored no championship points.
He returned for the Belgian Grand Prix at Spa-Francorchamps later that season, qualifying twelfth and completing the race in ninth place — a classified finish at a demanding circuit and a more complete performance than his Silverstone retirement. In addition to these two championship entries, Crossley took the Alta to a number of non-championship Formula One events during the period.
At the conclusion of the 1950 season, Crossley retired from front-line racing, citing the mounting and ever-increasing cost of staying competitive — a burden that forced many British privateer drivers out of the sport as the championship rapidly escalated in technical and financial demands.
A return attempt came in 1955, when Crossley built a new car — the Berkshire Special — powered by a Lea-Francis engine, and entered it for the Richmond Trophy at Goodwood Circuit. Practice revealed the car to be the slowest entrant on the circuit, and he withdrew without racing. Business commitments then prevented any further development, and he did not return to motorsport.
Crossley died on 7 January 2002 from a stroke at the John Radcliffe Hospital in Headington, Oxford. He was 80 years old.