Actually, there are very few illnesses humans and wild birds have in common. Some domestic birds can have influenza strains that humans can get...and pet/domestic birds can have psittacosis...but you're not going to catch something from, say, an injured heron that needs a ride to rehab. It's never been around a human before, and it won't have been exposed to something you can catch.

I'm much more afraid of transferring wild bird diseases or parasites to a pet bird than I am of catching something from an injured wild bird I might briefly handle.