To my knowledge cat litter is the big thing to avoid. I don't know about birds though. Maybe check with your doctor? We had a baby robin on our lawn that couldn't fly and was abandoned so I put it in a shoe box and took it to a friend that saves birds. I was careful not to touch it though because I didn't know if they carry anything harmful to my baby.
Birds are fine. Just use gloves when cleaning there cage as that's always a rule regardless of pregnancy. I have 2 African greys( one is 39, and the other 2) and 2 love birds. And yes let him/her do what they normally do