I don't think anyone has ever said you should have a permit in your hand within 60 days.
If you read the statute (N.J.S.A. 2C:58-4), you submit your application to the local chief of police (or local NJSP barracks if you don't have a PD or are from out of State). The investigation by that agency must be complete and your application denied for cause or forwarded to the County Superior Court for approval within 60 days. Some say the 60 day clock starts when you submit the application to the agency, some say the clock starts once your fingerprint results are delivered to the agency and your application is truly complete.
Once it arrives at the court, most are provided to the County Prosecutor's Office for review. Once the PO has no objection, the court typically issues the permit. If there are any hiccups, the judge can hold a hearing.
There is a JAD court directive #06-19 (not a law or reg) that judges should review permit to carry applications submitted at least once every 30 days. A PO should state any objections they have within 14 days, and a permit should be issued or denied with 30 days.
In theory, the process should take no more than 90 days, start to finish, notwithstanding "extraordinary circumstances."
While I am sure the "system" suspected this as coming, they weren't officially notified until June 24, 2022. As a result, the system, from the local PD to the Judge, had previously dealt with a small handful of permit applications - mostly from armored car guards and security officers who presented with a letter of need. I don't think it is unreasonable for these agencies to develop new systems to handle the influx of permit applications flooding the system. Be patient, calling and complaining is not going to get them processed any quicker - they know what they need to do.