What happens to snot when the nose is pierced?

Probably nothing. If what you mean by "snot" is mucus secretion from the nose, piercing of the lower nose cartilage should not have any effect on the quality or quantity of mucus secreted by the nasal glands inside the nose. The only remote relationship between piercing and nasal mucus is whether the location of the piercing produces an infection which could lead to the interior of the nose becoming infected.