How long does it normally take a nerve to die after a pulpotomy?

Hard to say: At least for baby teeth, the goal of a pulpotomy is to remove coronal damaged pulp tissue and for the remaining tissue in the roots to actually stay alive. If the remaining tissue does die, the pulpotomy has failed (a fairly common occurrence) and either removal of the remaining dead tissue is necessary (root canal therapy) or removal of the whole tooth (extraction) is appropriate, depending ..

That depends: The term "pulpotomy" refers to the partial removal of the pulp from inside a tooth. This is usually performed on children with developing teeth or adults as a quick emergency visit. Medication that is placed in the tooth should calm it down within days. If that is not the case you need to return to have more of the pulp removed "pulpectomy".