I have one that lets it run into her mouth, it SO grosses me out. And, lucky me, she is also the one that is always getting sick. I am lucky if she goes 2 weeks without a runny nose. I have kleenex readily available, but she could seem to care less.
But my younger daughter is the opposite and if she has any sort of feeling that her nose is running she has a fit and starts screaming that she needs a kleenex. It is a bit exhausting.
So I can't win either way! Lol
Just waiting for them to get a little bit older so they can both grow out of their totally opposite but related problems!
Don't have any solution for the nose picking, aside from keeping on them to not pick and to use a kleenex. If they were older and independent in thd bathroom I would send them to wash their hands every time, perhaps would be a deterent since they would have to keep stopping what they were doing to go wash hands.