I believe the Aus/NZ accent comes from London. It shares many features with Cockney. Regionally, the modern Kentish accent, which is very slight, has a faint echo of Aus/NZ, so I suspect Cockney (East London) and the more recent Saaf Lunnon derive from Kent, which is why they're so strikingly different from East Midlands (London and Oxford and Cambridge), the origin of RP.

The fact that the majority of settlers came from the major population centre of London would explain how Aus/NZ and for that matter South African, which in some ways is similar, came to be, without invoking class or convicts.

But it is striking that NZ is ethnically very Scottish, yet has no trace of Scots in the accent. A simple explanation in terms of regional origin of settlers obvously isn't quite enough.