Usually between 1-2 days and 1-2 weeks but could also be way shorter or longer. It depends on the quality of the request (how much is filled out and the sourcing), how obscure an object is (often exonumia are quite fringe objects with no catalogs, is there even a place for it already in the catalog etc.), does a referee for the relevant issuer exist (if there is no referee and a non specialized admin has to deal with the request point one would need to be of high quality to quicken things).
Appeals should not be handled by the referees but by Admins/Master Referees. There are few of them so it tends to take long before they bubble up to the top and probably needs conversations between the referee who rejected it and the one working the appeal. Still much better system than previously when one reject and out.