Searched Trove National Library of Australia , free ( Equivalent of NZ Papers Past ) for South Australia only and found just one article - From the S A Chronicle 26/12/1874 . John Jasper Wilson storekeeper , was fined fifteen shillings for "being too great a distance from his horse and dray" in Hindley Street.
You cannot search S.A. BDM online , but checked NSW BDM , no sign of John J or father Jasper .
On UK Free BDM there is a death of John Jasper Wilson, March Quarter 1903 aged 70 in Burnley, Lancs. Looked for death of Jasper Wilson , there are 4 from Sept QTR 1837 (when registration started) one in Dec 1859 , Leeds , before they gave age on index and the later 3 to 1900 are two babies and one 22 yo.
You said they went back to NZ , so if John Jasper started in NZ before Adelaide have you checked the Immigrant Shipping records