Car Insurance question – How to accelerate the process?

One week ago my car was struck form the rear while parked and pushed into another car. I have the police report that shows that It wasn't my fault, which I faxed to Geico (we both have Geico). I am trying to use the other party's policy, but Geico is telling me that they are still waiting for that person to call back.
They are saying that they have to wait for 30 days, before sending them a warning, etc.
I have collision coverage, but I didn't want to use it because my premium might magically go up?

My question is how can I get the ball rolling, it's obvious that the perpetrator is ignoring the insurance company calls. Should I just use my collision coverage, or is it worth it to wait?