If your insurance doesn't have to pay anything your payments probably won't go up.
I can tell you one thing though, don't trust the 66th precinct! I had a similar story in the neighborhood, but there was no one in sight besides for me and the other driver. I thought it was obvious that I was innocent, as I was parked, and that the marks on my car clearly indicated that the truck slammed into me, and not vise versa. When I picked up the police report, I was surprised to see a "witness" claiming that the truck drivers story, that I actually moved out of the parking spot and hit him.
So if you don't have someone who was on the scene, take a friend, and walk into the precinct, and tell them you want to update the police report with the "witness" that left the scene before the cops showed up. Don't let yourself get screwed over by these crooks in blue.