How can I get my three-year-old to stop picking his nose?
I don’t think you can. I would suggest you mention this to his/her family doctor to check his nose to see that there is nothing wrong.
You can insist that your child not nose-pick during meals, or maybe even in public. If you want to do this tell him it is not polite and that he should do it in private. Have him leave the table if he starts. Praise him for polite behavior.
Trying to control nose picking all of the time is doomed to failure. Focus on other goals.