HI there, this is Dr. Elaine
It is possible that the puncture went all the way from the outside into the nasal cavity/sinus.
There are probably dried bits of blood/clots within the nasal cavity that she is trying to expel, and as the hours have passed, inflammation/swelling is likely present, as well as some irritation. The blood inside the nasal cavity has likely dried, and inflammation can make her feel as though she has a stuffy nose.
Sometimes dogs will do something called a reverse sneeze, whereby they are trying to get fluid/material from the nasal cavity into the back of the throat. there are varying degrees of intensity to this, and such a young puppy likely would not/could not do this action with the same intensity as an older or larger dog could. See the following:
may or may not be occurring. The biggest concern I would have is for infection to develop in the puncture wound(s), so your veterinarian may want to prescribe antibiotics (usually liquid for young small patients). While this may not be an emergency, I would consider having her seen sooner rather than later since any dog bite wounds have the potential for infection. If bleeding begins, cold packs gently held on the top of the nose can help, but obviously if bleeding is severe or starts up all of a sudden, you must have her seen.
Hope this helps.