Depends on how given. If given intravenously, it works literally in seconds (10-20 seconds to start working, works fully in couple minutes); if given intramuscularly (im) it takes 2-10 minutes to kick in. If given in a nasal spray (which is being used in several communitites to help addicts and their friends treat od) it works in-between IV and im, starting in 1-2 minutes, and taking full effect by 5-10 minutes.
Administered how? “Naloxone has an onset of action within 1- 2 minutes following IV administration & within 2 -5 minutes following subcutaneous or IM administration.” REF: https://www.fda.gov/downloads/advisorycommittees/committeesmeetingmaterials/drugs/anestheticandanalgesicdrugproductsadvisorycommittee/ucm522690.pdf . Intranasal is similar to intramuscular.