How long should insurance take to fix my vehicle?

I live in CA, another guy hit my car while I was completely stopped, a couple police officer witness the accident and made the report, the other guy obviously was at fault.

It’s been 3 weeks and I haven’t heard anything from the insurance or the guy that hit my vehicle, I made a claim with his insurance and my insurance too, I even got an estimate on the repairs, should I just take my vehicle to the shop and have my insurance deal with the other insurance? If I get a rental car during this time can I charge the other insurance for it? Or should I just wait longer? Is there a time limit frame for the insurance to respond? Thanks in advance.