NZ would be better avoiding any UK link at all, dealing direct with OMT and acquiring a licence to the Iver Huitfeldt design and having the ships built in a South Korean yard, fitted out to NZ specs. Would work out cheaper in the long run. The original ships were built to military specs but built using commercial practices in the OMT yard (now defunct) The design also includes commercial practices which makes the Iver Huitfeldt FFG class easier and cheaper to operate, maintain and upgrade.
Politically the RAN build would be the one to watch because some will see NZ acquisition of Australian built ships as being part of the AU-NZ defence relationship and wider relationship between the countries. Canberra would see such acquisition as NZ taking defence AND the AU - NZ relationship seriously IF NZ acquired three or more frigates without any reduction of the ships capabilities. However NZ would have to basically sign cast iron guarantees that it would stick to its agreements and if I were Canberra I would ensure that there was a very hefty financial penalty for any Kiwi pollie back tracking - AU$1 billion as a start point comes to mind.
NZ would also be able to take advantage of the RAN continual build program meaning that both countries would be in lockstep. The advantage of this is that a minimum of 12 ships would be built and build costs really come down from ship nine onwards. The ANZAC build was 10 and they were built on time, under budget, with ships 9 and 10 very cheaply. The funding that would normally be spent on a MLU could be put towards the replacement so that is the advantage of a continual build.
The USN FFX is another option as well. If the USN go with the T26 then that adds to the RN / RAN / RCN group and I think that the NZG would have to serious give a T26 variant very serious consideration because basically it will be a FEYES frigate. It's two thirds of the way there now and the P-8 is is now basically the FEYES MPA / MMA. In NZG terms the FFG have the same same status as the P-3K2 / P-8 - strategic national assets, so if three or four FEYES partner navies have a T26 variant, that would carry serious weight with Wellington.