Cordts was born in Hamburg and emigrated with his family to Sweden at the age of two. He developed an early and deep interest in mechanical equipment — working on farm tractors as a child in Sweden and riding motorcycles as a teenager. In 1954, when he was approximately 18 years old, he emigrated again, settling in North Bay, Ontario, Canada, describing the move as the best decision of his life. He raced motorcycles in Sweden before his arrival in Canada.
Cordts began car racing in 1956 after purchasing an MGA, competing initially on ice on Lake Nipissing and then moving to circuit racing at Green Acres and Harewood Acres. He drove through a succession of amateur and club events in MGAs, an Austin-Healey 100-6, and an Elva Courier, recording numerous wins before catching the attention of established figures in Canadian motorsport.
A decisive step came in 1965 when Dave Billes offered him a drive in a Performance Engineering Chevrolet Corvette. Cordts won the Canadian Championship for big bore sports cars that season against strong competition that included the Comstock Ford Racing Team, and was voted driver of the year by the Canadian Race Communications Association. Billes then gave him a succession of McLarens, beginning a long working relationship that became the foundation of Cordts's professional career.
Cordts held lap records at several circuits. His most noted achievement in this regard was setting a track record of 101.8 mph at Harewood Acres in 1968 while driving the Performance Engineering McLaren Mark IIIB — a record that stood until the circuit closed in 1970. He also held lap records at Portage la Prairie in Manitoba, Watkins Glen in New York, and Westwood in British Columbia.
Cordts became a regular in the Can-Am Series from 1966 onwards, competing primarily in McLarens and Lolas provided by various teams including Performance Engineering, Young American Racing, W.O.R.L.D. Racing, and U.S. Racing. His Can-Am schedule took him to circuits across North America including Mosport, Watkins Glen, Road America, Riverside, Bridgehampton, Laguna Seca, and Westwood, as well as Mount Fuji in Japan. His best Can-Am finish was second place at Road America in 1974 with the Kaplan team's McLaren M8F.
In 1966 he recorded two wins at the Mosport Spring Trophy in a McLaren M2B, and in 1967 he finished third in the US Road Racing Championship standings for Can-Am eligible cars over two litres. He also drove Roger McCaig's Formula 5000 McLaren in later seasons, recording a second-place finish at Road America (Elkhart Lake) in Formula 5000 competition.
Cordts made a single Formula One World Championship appearance at the 1969 Canadian Grand Prix at Mosport on 20 September 1969. He drove a Brabham BT23B fitted with a Climax engine, entered by Paul Seitz. Starting from nineteenth on the grid, he retired after ten laps with an oil leak while running in sixteenth position. Despite the brevity of the outing, the appearance placed him among a select group of Canadian drivers to have started a Formula One World Championship event.
Cordts competed in the SCCA Trans-Am Series, where his most notable involvement came with the BF Goodrich-sponsored Pontiac Firebird operated by T/G Racing, a car widely known as the Tirebird. BF Goodrich used the programme to prove the race credentials of its Radial T/A tyre — at the time the first American DOT-rated tyre with SCCA approval for professional competition. Cordts drove the Firebird in Trans-Am events alongside the Sebring 12 Hours and the Daytona 24 Hours. In the car's first Trans-Am race at Mosport he finished third, and the Firebird subsequently scored a class win at Watkins Glen, reported as the first production car to win a race using radial tyres.
He also drove FIA events with the John Greenwood Corvette team alongside Don Yenko, competing at the Daytona 24 Hours and the Sebring 12 Hours.
Cordts maintained a deliberately low public profile throughout his career. He was described by journalist Frank Orr of the Toronto Star as the only racing hermit driver he had encountered — an impression reinforced by the fact that, at one period, the only way to reach him was to leave a message at the local general store in Port Sydney, Muskoka, where he lived. His own account of his early career — living in a tent behind the pits, surviving on turnips and rabbits, funded by prize money and the generosity of friends — became the basis for his 2006 autobiography, Blood, Sweat and Turnips. Road and Track magazine wrote in 1970: "If I was the owner of a good racing car, I would want John Cordts to drive it."
After retiring from racing following the end of the Can-Am in the mid-1970s, Cordts settled on Vancouver Island in Western Canada, where he became known as an accomplished wood carver and sculptor. He was inducted into the Canadian Motorsport Hall of Fame in 2003 and into the North Bay Sports Hall of Fame in 2013.