Oatey, the son of South Australian football great Jack Oatey, captained the first eighteen at Norwood Boy's High School winning the Hone Medal for Fairest and Best in the High School competition in 1959 and 1960. In 1960 Robert was invited to join the Norwood thirds (U19) for 4 games as a Reserve after playing for his school. He then played in the SANFL second semi and Grand Final which Norwood won by defeating North Adelaide FC 13 goals to one with Robert winning the prize of $10 as best player on the ground.[3]
A rover, Oatey kicked a goal with his first kick in league football for Norwood, in the 1961 season.[4] He was amongst Norwood's best players in the loss to West Adelaide in the 1961 SANFL Grand Final.[5]
Oatey won Norwood's best and fairest award for the first time in 1967 and was also the club's leading goalkicker.[6] The following season he began a six-year tenure as Norwood's captain-coach, during which time he played his best football.[3] He was runner up to Barrie Robran in the 1968 Magarey Medal count, in his first year as coach, a season when Norwood finished last on the ladder.[4] He won another club best and fairest that year, then again in 1971 and 1972. Under Oatey, Norwood improved to fifth in 1970 and were fourth in both 1972 and 1973, but he would be replaced as coach by former North Adelaide player Bob Hammond in 1974.[7]
From 1974 to 1978, Oatey played for Sturt, which were coached by his father.[3][6] He was a Sturt premiership player in 1974, when the club defeated Glenelg in the grand final.[3]
He received the Medal of the Order of Australia (OAM) in 2008, for his "service to Australian Rules football as a coach and as a contributor to the development of younger players."[12]
Victoria (VFL) 16.23 (119) defeated South Australia 7.9 (51), at North Hobart Oval, 11 June 1966, crowd: 23,764 Western Australia 13.11 (89) defeated South Australia 10.14 (74), at North Hobart Oval, 13 June 1966, crowd: 13,969 South Australia 21.20 (146) defeated Victoria (VFA) 9.11 (65), at North Hobart Oval, 16 June 1966, crowd: 10,199 South Australia 14.7 (91) defeated Tasmania 9.13 (67), at North Hobart Oval, 18 June 1966, crowd: 23,368
