For uninsured motorist claims, it's likely that your insurer would want a police report of the incident. What I don't know would be if it would be possible to still get a police report and redo the claim.
Unless you were under the influence, drove recklessly, acted with intent, or fled the scene in a hit-and-run, I don't necessarily think you could be charged for a felony if you've been cooperative. However, a suit could be filed against you for damages.
Even if the vehicle you backed into was uninsured, it is still your responsibility to pay for the damages if you were at-fault. If the other party doesn't want to go through insurance, then they don't have to.
Unfortunately, you and the vehicle were not insured during the time of the accident, and if no policy was bound at that time, then you would not have coverage for the accident. Insurance will never backdate and trying to get them to cover an accident that already occurred is considered fraud.