Despite in the same alliance, SQ/NZ partnership is a bit nah - poor mile earn rate with SQ on AIr New Zealand flights (especially in Y, but J looks alright *1), close to zero reward flight availability.
CX and NZ are joint venture for AKL-HKG, so you can earn Cathay's Asia Miles or NZ's Airpoints on direct flights irrespective to the metal - but accruing to other OW/SA FFPs will not be as straight-forward.

I would keep the Avios account for AKL-HKG and QF/OW flights since Avios can be freely transferred between QR and BA, though will compare the earn rate between CX, BA and QR. But CX is another odd one when crediting to another FFP *2.
Although there are quite some issues CX business class, the reverse herringbone on CX is still better than the herringbone business class offered by NZ.
Using Amex global transfer for a membership rewards card might be an option.

*1 SQ partner earn chart https://www.singaporeair.com/saar5/p...FOALLevels.pdf
*2 CX hiding fare classes, QR BA earn rates are different Trip.com and Cathay Fare Codes?