Green’s Dictionary of Slang

(orig. milit., a doctor; thus S.Afr.) a prison doctor.

1925		Fraser & Gibbons Soldier and Sailor Words 296: Vet, the, medical officer.
1938		Partridge World of Words 196: Doctor becomes [...] the vet or [...] pill-shooter.
1975		A. Powell Hearing Distant Harmonies 83: Saw my vet last week. Said he’d never inspected a fitter man of my age.