Congratulations!
How long have you been living with your partner? - that you can prove with solid evidence, I mean.
That aside, it's very important that as of NOW, you should collect every single thing you can think of that proves you are living together. Here is an INZ list, and old threads on here, to give you the idea of what is necessary.
http://www.immigration.govt.nz/migra...artnership.htm https://www.google.co.uk/#q=site:enz...tnership+proof
What I am thinking, and/but you should confirm this by talking to INZ, is that you could apply for a partner-sponsored temporary work visa.
http://www.immigration.govt.nz/migra...milystream.htm They may not be able to complete processing the application without the X-ray, but it would be able to be completed after the birth of the baby. Once you have an application in for a visa, you will be granted an interim visa
http://www.immigration.govt.nz/migra...erimvisaqa.htm to keep you legal, if the processing can't be completed before your previous visa runs out.
Your baby will, if born in NZ, be born a NZ citizen, and that, and the fact that your partner is a NZ citizen, makes you eligible for full use of the NZ maternity and neo-natal care services for free.
http://www.health.govt.nz/new-zealan...nancy-services