As your question is about BA miles & BA FF program you would get a better answer on the BA forum. A lot of BA FT'ers will have used BA miles to get downunder.
For QF the fare buckets are in this
old link by
Dave Noble. (A link in the QF forum sticky
The ULTIMATE Qantas Guide ) Not sure it they are all still correct. I think some were changed last year. For BA buckets see the BA forum. For CX buckets see the CX forum etc
To get to AKL you can go
- Via Asia on BA to Aust & then QF to NZ
- QF all the way via Asia
- BA to the USA and the QF LAX-AKL
Via Japan or Hong Kong are other options
However the chances of getting a reward ticket, in any class, each side of Xmas - New Year to NZ-Australia are close to zero or less. Peak travel for us in Au. QF rewards for this time get snapped up as soon as QF release them (353 ? days)
Time to start looking for a paid ticket IMHO
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Edit:
Found the BA buckets on the BA forum ^ (where they belong of course

)
A link in the BA Forum sticky
READ THIS FIRST >>> The Ultimate BA Guide! to
Fare ‘Buckets’/Letters
Edit 2:
AA buckets are in the
AA wiki. A link from the
Unofficial Users' Guide - American AAdvantage Forum