Initially I thought you may have been planning a big roadtrip. If that's not the case then it doesn't matter much which order you travel.
Stinger season is from late October typically, just wear a stinger suit in the water. Cyclones can hit the North from November onwards. So maybe go north early in the trip.
There are direct flights from Cairns to Auckland (Qantas and maybe Air NZ) but a lot more choice from Sydney or Melbourne.
As I said we did 3 weeks NZ NI this year and 3 weeks SI last year and we have previous NZ trips.
A lot to see on both islands.
Our SI trip was focused specifically on the Southern Alps and Fiordland. We flew in and out of Christchurch. Do the Alps and Fjords interest you? The scenery was to die for. Just extraordinary!
Our NI trip was more of a mixture. We flew in and out of Auckland (from Melbourne) and did a sort of figure 8 heading south then back through Auckland and then headed north as far as the Bay of Islands.
Without knowing your interests it is hard to advise what to do with such a tight schedule.
I did post some pretty crappy trip reports on both little adventures so that might provide some inspiration.