How long does a police investigation take for a minor hit and run?

The police usually won't put much time investigating a small fender bender. Report it to your insurance company to resolve.

My experience is that most police departments do not have the resources to investigate minor collisions. You should report the incident to your insurance carrier. If you are able to locate the other driver, you would have your witness to support your position.

How much effort the police will put into an investigation like this will depend on the jurisdiction involved. In Denver it is likely that little more than taking a report will be done and no further investigation will be attempted. If you suffered damages and can identify the other party, you can file a civil lawsuit. If your damages are less than $7,500 you can file in small claims court.