First of all, make sure whether your current employer may fire you if you work for another employer. Not sure about the UK, but in Germany, you always have to file it with your employer, who may object to the side job of a full-time employee for various reasons. (E.g. you have to be fit when you work for your "main" employer, so you may not work for another employer in your "recreational time" without his approval.) Also make sure that you don't break any employee protection laws (maximum hours per day/week, minimum pause, working on Sunday/Holidays).
Second, check whether the "side job" is you being "self-employed" or whether you are legally a normal employee of the NZ company. If the first, make sure you don't break self-employment laws in the UK (which you may, by having only one client); if the second, make sure that you don't break employment/tax laws in NZ (which you may, not having a NZ work visa and/or paying taxes in the UK only).
If you got there successfully, you will then be on your own with your taxes. Your company accountant will do the (pre)tax as before, but you have to make sure that you declare everything you earn and pay everything you owe. Visit a tax adviser specialized in taxation of international work contracts; I would also recommend to contact a NZ-based tax adviser as well.