Hi, I worked in a petshop until recently, plus I have several animals at home here (as you do when you work with animals). We also live on a farm. The only animals strictly off limits were kittens (for the toxo) and sheep and lambs during lambing season.
I have handled every other type of animal while pregnant and have to agree with mostly everyone here, it's about hygiene. I keep hand sanitiser beside my bird cage and use it after handling them, plus I have gloves, and routinely wash my hands. I wouldn't handle the birds or be too stringent with cleaning the cage if you have an open wound on your hand however, as bacteria can easily get in (this is when those kitchen gloves come in handy). Other than that you should be fine.
Budgies are great as they are cheap and can be trained to do tricks and talk, are small and don't make too much mess. I think they have real personality but then I like all birds.