Check out the Royal New Zealand Aero Club website aka Flying New Zealand.
Here's a link to a map showing where their member clubs are.
CLUB LOCATIONS
Have a look and decide where you want to spend your time and choose a club near by. Some clubs operate weekends only some are full time, the standards are generally very good though like anything can vary. If you post where you wish to reside while in New Zealand then I could provide some suggestions on which may be your best options.
So far as I know there is no one that does JAA licences here in New Zealand for people such as yourself, your best option is to do the New Zealand PPL and convert to a UK or JAA licence on your return
So far as I know you will need to do the exams here in New Zealand though most clubs can set you up with the study material so that you can self study prior to arrival to speed up the exam process once you arrive.
You will need to do a medical. Have a look here to find a Doctor that can do these for you in the UK
Directory of Overseas Medical Examiners
I know of several people from the UK who have done what you intend to do and they have easily converted their licences on arrival back in the UK.