Hi Sadie,
Usually budgies and cockatiels can have the same type of cage. If the budgies can get their faces through the bars as far as their eyes or more, then the bars are too far apart. Mesh is not a good idea – most of it is made with lead weld and would be toxic for them, and rigging something like that is dangerous because they can get toenails or their beaks caught in it. If the bars are too far apart, you need to buy the right kind of cage.
If you have both pairs in one cage, you can’t give them nest boxes. You can only have one pair per cage when you breed birds. Otherwise they will fight, often to the death, and they will also raid each other’s nests and destroy the eggs or kill the chicks. So go ahead and remove the nest boxes if the pairs are together, regardless of what the seller told you. Most likely, these birds are not good breeders, if they have actually bred at all. Nobody sells good breeding pairs, unless they are getting out of the business entirely. It’s too hard to get a good pair that produces chicks without there being some issue, so you can see why it’s not likely these pairs are actually breeding pairs. Sometimes a breeder will claim to be downsizing, and the pairs they sell will have some reason why they have never produced healthy chicks. So you can get another cage and set both pairs up, but my guess is they will be infertile, or be egg eaters, or not care for chicks. I would just enjoy them as pets, get them some fun toys and watch them interact and play. It’s a lot more fun, a lot less work, and a lot less expensive to keep them as pets than breeders anyway. Breeding means extra foods, a special place for the breeding cages, extra cages in case they hatch and raise chicks, and generally a lot of disappointments. You have to remove the nest boxes after every clutch of eggs, whether they hatch or not, and make the pairs rest for 6 months. You have to limit them to only two clutches per year, or it puts the female’s health at risk. And at most they have 3-4 years they can breed before they are too old. You also have to be able to find homes for any chicks, or have extra cages where you can house the males and females separately, because you can’t leave them with the parents once they are weaned, and you can’t let related birds breed.
Budgies are a lot of fun, so again, just enjoy these birds as pets.
Thank you for asking Lafeber,
Brenda