Some unethical web designers and other registrars fail to set you as the registrant name, and instead use their own name. This is against the .nz rules.
If you have paid for the domain name, the registrar is required to have registered it under your name and provide your UDAI within 1 business day.
If they don't do that, you can make a complaint here: http://dnc.org.nz/story/drs-home
Sorry there is nothing more we can do to help, except for you to sort through the dispute with your old domain provider.
In the meanwhile, you could register an alternative domain, or ask that the old web designer or registrar update your nameservers to point to ours so that you can at least get your new website live.
As soon as you have your UDAI, we can transfer your domain and start helping you. Without that key, we have no power to help, and we can't get involved in your dispute.