What can I expect for a car accident for which no fault was established but the vehicle that I was driving was uninsured?

The first option is to file a claim with the other person's insurance carrier to go on the offensive to try to get the other driver's insurance company to pay for your damages.  the insurance company is going to do an investigation and rely mostly on their insured as to what happened - who is bias.  The insurance company  is likely going to deny your claim becasue there is no police report and because it is a he said she said claim.  Therefore, you will need to file a lawsuit in court to prove the liability and the damages.  The other option is to do nothing.  However, the other driver may try to make a claim on your insurance carrier, which may result in a rate increase.  The other driver may do to you what I suggested that you do above (file a claim).  The other option is the work this out with the other driver without going through insurance - that may be the best option.