"According to UK licensing" that is correct - it is all SEP with differences training signatures for tailwheel etc.etc. as you say.
However, for NZ licences things are different! Aircraft-specific "type ratings" are required from the word go. For each aircraft you fly you have to get a "type rating" - whether it's for a PA28s, C150, C172, PA18 whatever.
For a NZ PPL holder flying in the UK - as you would be flying under the privileges of your NZ licence, then the NZ licensing rules are the rules that apply.
You can't fly any other aircraft unless you get a "type rating" signed off by a NZ instructor, and post the forms back to the NZ CAA.
easy as that.