NZ is certified rabies free and is a part of some of the Euro exclusions to quarantines, check if rabbits are covered, dogs/cats do not need to be quarantined if they meet the travel standards. http://www.biosecurity.govt.nz/exports/animals/uk-pets-scheme.htm – Also, UK allows some animals to be quarantined at home, if they are ones that typically never leave the house. A friend was able to take his bird to the UK from the US and they let him do at-home quarantine so he didn’t have to leave his parrot in a facility since it would stress him out too badly.
From NZ to the U.S. you can bring animals with just a health certificate, since NZ is classified rabies free, EXCEPT to Hawaii. Hawaii has strict import laws due to the fragile ecosystem of the island. But, given how much it costs to live there, probably don’t wanna move there anyways! lol
http://www.biosecurity.govt.nz/going/personal/pets <– there is a site with some info about taking pets to various places. It focuses on cats/dogs though, so you'll need to call for info on rabbits.