[1] Veitch was born at Dunedin in 1937 and educated at Otago Boys' High School in the city.
[5] From the 1953–54 season Veitch was selected in age-group sides for Otago, including playing in the Brabin Shield competition.
[6] He scored 55 runs for the Otago under-20 side against Wellington in 1953–54, "shaping well" in his innings,[7] before scoring a "well compiled" 65 against Auckland later in the season[8] and at the beginning of the 1954–55 Plunket Shield season Veitch's name was proposed as a possibility for the senior Otago representative team whilst still at school.
Opening the batting, he made scores of 27―an innings called "a classic" by The Press[13]―and 40 runs and was retained in the Otago team throughout their victorious Plunket Shield season.
[6][14] Described as a "forceful" left-handed batsman,[15] Veitch made four appearances for Otago during 1958–59 but dropped out of the team for the following season and played only four matches for the province between 1960–61 and 1963–64.