Born in Leeds, West Riding of Yorkshire in 1948, Harvey attended Foxwood School, and played for Leeds City Boys before leaving school to work in a Styloshoe factory. He was signed as a professional footballer by Don Revie in February 1965, after having played as an apprentice for two years. Harvey was the reserve goalkeeper for Leeds United during the late 1960s and early 1970s, and was on the reserves' bench when Leeds won the League Cup and the Fairs Cup in 1968, the League championship in 1969, and the Fairs Cup in 1971. In the 1970 FA Cup Final, an error by first-choice goalkeeper Gary Sprake allowed opponents Chelsea to equalise in the first half. Sprake was blameless for the second equaliser, but Harvey replaced him for the replay, which Leeds lost 2–1. In the 1971–72 season, Harvey played only eleven matches, but was selected ahead of Sprake for the 1972 FA Cup Final against Arsenal, as Sprake was suffering from a knee injury. Leeds won 1–0 with an Allan Clarke goal, and from that point on Harvey was the first-choice goalkeeper. Harvey played 63 times in the 1972–73 season, and played in both the FA Cup Final and the European Cup Winners Cup Final – ending up on the losing side both times. Leeds put together a 29-match unbeaten start to the 1973–74 season, winning the League championship and earning Harvey the title medal to which he had not been entitled five seasons earlier. At the start of the 1974–75 season, Harvey took the last penalty during the shoot-out at the Charity Shield game against Liverpool but missed, thus allowing Liverpool to carry the trophy home. Later in the season Harvey was injured in a car crash. He missed the rest of the season replaced by his understudy David Stewart. Stewart was praised for his bravery and "inspirational saves" beating Anderlecht and F.C. Barcelona in the 1974-75 European Cup. Leeds were beaten 2–0 by Bayern Munich in the final in Paris. The rest of the 1970s saw Leeds decline after the ageing team Revie built broke up. Harvey left in 1980.
Harvey moved to NASL club Vancouver Whitecaps in 1980. He had an unsteady first season with Vancouver, and even though his form improved in the second season in the NASL, injuries from another car accident prevented him from regaining his form and place.
Return to Leeds United
Harvey returned to Leeds in 1983, by which time the club had been relegated to the old Second Division. By the time he left in 1985, he had played under three of his old teammates: Allan Clarke, Eddie Gray and Billy Bremner.
Harvey made his debut for Scotland in a 2–0 win over Denmark, selected through his Scots father. In his 16 full caps he was part of eight victories, four draws and four defeats. He was selected as Scotland's first-choice keeper for the 1974 FIFA World Cup in West Germany. Scotland went out after the group stage unbeaten on goal difference after a 2–0 win against Zaire and draws with Brazil and Yugoslavia.
Career after football
He retired from football at the age of 37 years. He managed a public house at Stamford Bridge, near York. He then became a postman, saying, "The sorting office atmosphere was like a dressing room". In 1994, with his wife, June, he bought a 150-year-old stone cottage with 10 acres of farmland on Sanday in Scotland's Orkney Islands. He again worked as a postman. On 24 December 2009, he suffered a heart attack, from which he recovered.