Poseidon. Welcome to the forum.
I wouldn't worry too much about gathering a lot of things together to prove your relationship. Probably just 4 items will be enough as INZ won't have the time to go through a lot of paperwork.
If you have been living with your partner for 12+ months then an application under the
Partnership Policy should be fairly straight forward and you wouldn't need to look at any other visa types.
You should apply for your visa at your nearest branch of INZ but I don't think anyone in the NZ branch would be able to see what's going on at the London branch. They can send information between the 2 branches if it is requested but that would be all. If you want to apply from NZ you would be able to apply for a
Work Permit under the Partnership Policy which will give you enough time for your residency application to go through. I'm not sure how long you would be allowed to say in NZ on a tourist visa from your own country so the WP route would be another way of hanging around until the application is finalised.