Re couping loss from an accident not my fault

What is the name of your state (only U.S. law)?
Tennessee.

I was in an accident last Friday in which the other party was clearly at fault. At the time our cars were drivable and the other party had insurance so I did not contact the police foolishly. I have filed a claim with his insurance company but he has not responded. His insurance company said that they will not do anything without the other party answering the claim, in essence the claim will be dropped if he chooses not to respond. I was wondering what legal recourse that I have either through the insurance company or through small claims court to have my issue heard. I understand that at this point it is my word against his, but without any response it seems that he is able to avoid the situation all together. Do I have any recourse after not calling the police? This happened in front of a day care center in which there were witnesses.

Your choices are to file a claim with your insurance (if you have collision) and let them pursue the other party, or wait for the other party's insurance to complete their investigation. You COULD file a small claims suit against the driver, but IMO it's a little early to start that. Investigations can take time especially if the driver is hard to get ahold of.

Do you have contact information for the witnesses?

Go down to the local police and report the accident at your soonest.