Hi, I had to go through the same thing when my permit issued by Phila county expired during last year's shutdowns. Your timeframe is a bit longer than mine, I applied online* earlier this year and got my appointment scheduled in around two months. So I'd say your quote of three months is not out of line, although a bit longer.
*: after they canceled my appointment I booked in fall 2020 WITHOUT any notice to me, I might add.
Anyway, I just picked up my permit and if this is your first time at the new location on Erie Avenue it's nowhere as easy to access as the old location on 10th and Spring Garden. Avoid driving north up Broad Street if possible, too many lights and you'll go through some not so nice parts of the city. Also, I had no trouble finding parking on the other side of the street but you can always use St. Christopher's Hospital's indoor (pay) lot as a backup, it is not far away.
Best of luck.