Getting angry and/or punishing her will not work. She's doing a normal thing for her age. Ignoring it doesn't make it go away either. You've received several good suggestions.
I'll add to those by saying she may be picking her nose because it's dry, itchy, or just stuffy because of buggers. And as one mother said, "what does a 3 yo do when there's a bugger on their finger?" So I believe it's a matter of education and prevention in a calm matter of fact way. A few "that's gross" comments in an offhand or funny way may be included.
I still have to put vaseline in my nose during the winter because my nose dries out and buggers build up. I often dig them out, because it's faster, in the morning after using vaseline. Of course I don't eat them. tehe But I did as a kid.
I suggest putting vaseline in her nose or she can do it herself. A part of baby hygiene is to wipe their noses out with a Q-tip. You can use vaseline in the nose of a baby also. So you are just teaching her how to do her own hygienic care.
I would not say that eating buggers could make her ill. They won't. The bug is already in her stomach if the buggars are caused by a bug. If she doesn't have a cold or respiratory infection the buggers are just dried out mucous. However, she can get an eye infection, if her buggers contain a germ, and she touches her eyes. I get frequent eye infections now because my aging eyes are very dry. I didn't get infections as a child in part because the tears wash the germs out. If she hasn't had an eye infection up until now I wouldn't worry about it nor would I suggest it to her.
So.....eating buggers is primarily an etiquette issue. My grandchildren eat buggers at home. I don't think they try to hide it. Their mother and I just don't pay attention to it very often. They don't pick their nose away from home which is better than I can say for myself. If I have a dry, itchy, plugged up nose I sometimes find myself scratching inside of it while holding a tissue in front of it because I cannot leave the area. Yes, that's gross! even with a tissue. I cannot stand an itchy, dry, plugged up nose. It will distract me from whatever I need to be focused upon. :):) if adults still do it, be gentle while teaching children to not do it. At least the adults know better.