Regarding the bird species you mentioned, I do not know anything about kakarikis, but I have looked at the other ones.
Parakeets and zebra finches can do well together in a large aviary. Zebras are a fairly sturdy and assertive/semi-aggressive finch. This serves them well when dealing with parakeets. They are still small and vulnerable to attack, if you have an aggressive budgie, but they usually do fine together, so long as there is enough space for both types of bird to do their own thing.
Both cockatiels and bourke's parakeets are less assertive and quieter than budgies. They can get bullied or harassed by the more aggressive and bossy budgies. Some birds will tolerate this better than others. And some budgies are better at "reading the crowd" than other budgies. It might work out or it might not - a lot depends on your birds, your aviary space, and the established flock dynamic.
Cockatiels are larger and dustier than parakeets and finches. They generally don't get bitey unless they are really pushed ... but budgies and zebra finches can get pretty pushy. You'll want to watch how they interact and make sure the cockatiels are doing okay with their roudier neighbors when they move into the aviary. But since you already have the cockatiels, I assume you should have a pretty good idea if they are able to tolerate each other. Just make sure that you provide lots of extra space and additional feeding stations so the birds can all attend to their needs without extra stress from competition.
From what I've read, Bourke's parakeets should not be a problem for the cockatiels or anyone else. They are about the same size as a budgie but with a calmer personality. They don't climb around as much as budgies, preferring to fly instead. And they are more active in twilight hours than other parrots, which can sometimes disturb other sleeping birds in the aviary. Overall, they are a nice little parrot. I don't get the impression that they would start trouble with anyone. That will be the budgies, primarily.
Overall, I recommend caution when mixing birds.
Research each type of bird thoroughly. Quarantine any new arrivals away from the main aviary for at least four weeks and get a vet check. Make sure they are healthy before introducing them to your flock. Monitor interactions and be prepared to separate birds, if there are signs of aggression or stress. Watch out for birds that are unable to access water or food due to bullying or over-crowding. If you are mixing species, aim to significantly under-stock your aviary to reduce stress.
Keep in mind that your birds are going to be happiest with more of their own kind. The addition of other species are unlikely to improve their quality of life and might have a negative impact instead. Proceed with caution and consideration for your current flock.