Hope you get some answers.
Depending on the density of a foreign object and it's location, it may or may not show up on an xray. This is especially true if there is swollen tissue. Have they put him to sleep and done a thorough probing exam? Is there a nasal discharge? Does he scratch excessively?
Like my vet says - "Every case with a bird is a research project."
I just noticed that you have two cockatoos. Are you aware that macaws are very sensitive to cockatoo dander and will develop a disease called "Pulmonary Hypersensitivity Syndrone" over time? http://www.thebirdclinic.com/uploadedfiles/pulhyper2009.pdf . There is no cure, but a macaw's life can be extended and their quality of life can be improved.
Good luck and please let us know what they find so we can use it for future reference.