I'm the victim of a hit and run.how long does it usually take a claims adjuster to investigate the claim

Depending upon the nature of your claim you should have an attorney represent you.

There is no set time for the adjuster to investigate your claim. But if you don't have a lawyer, I would suspect that the insurance company will drag their feet since they have nothing to fear from a unrepresented claimant.

This should be handled almost immediately. I assume you are proceeding under your UM / UIM coverage which makes this a first party case. Your insurance company should bend over backwards to cooperate with you. See an attorney if you feel that they have not met this requirement.

The timeline varies depending on the amount of damages involved. The claims adjuster will very likely drag his/her feet if you do not have an attorney working for you.