The target for NZ is Rattus Rattus and the Brushtail possum, they are the main predators that raid nests for both eggs and nestlings. The technique has so far been tested in mice but they are not the first concern. Stoats, weasels, feral cats are all issues in NZ but the rats and possums are the big targets since they don't just go after adult birds. Hedgehogs have been seen raiding eggs, but they prefer slugs and the population is small so they might be able to live alongside the native species with no major impact.
Another big problem is actually German Wasps, they eat the honeydew that the birds need, significantly harming breeding success.