09R, here's a good start to look at flying clubs, the
Flying NZ website (formerly known as the Royal New Zealand Aero Club). The link has a list of the affiliated clubs throughout NZ where you may be able to hire aircraft and/or complete your flying requirements for a Temporary Validation Certificate.
What SL has said about what to bring is correct, the only thing I recommend on top of that is contacting the club you wish to use prior to departing the UK to go over your holiday plans so that they can determine if they have the resources to meet your needs. Some Aero Clubs in NZ are quite small and may not be able to cope with the absence of one plane for a few weeks.
cavortingcheetah, Sea Bee Air closed its doors around 25 years ago.
