The process involves completing a NZ BFR, as previously stated, and then submitting the paper work through to NZCAA for the issue of a PPL Licence. This can take anywhere from 2 - 6 weeks to complete. An easier option may be to request a short term flight permit based on your UK PPL, this will give you NZ PPL priviledges on your UK PPL, but for a limited time frame only.
The chap to speak to is John Parker at CAA, www.caa.govt.nz
and he can give you the process to follow. This also takes time to organise, so I suggest you get in contact with him ASAP.
As for the hire of aircraft, most regions in NZ as well supported by aeroclubs, and will require you to perform a dual check prior to hiring you an aircraft.
Any other questions, please don't hestate to PM me.