Legal Question in Personal Injury in New York

It depends entirely upon the municipality. Many of them are represented by insurance company lawyers, who may or may not move quickly. In any event, you need to pay attention to the relevant statute of limitations that applies to your underlying claim, and not just the 90 day period in which you had to file your Notice of Claim.

If you are not repreented by an attorney, you should get one. Trying to litigate pro se against a government body or agency is unlikely to lead to anything but frustration.