In UK (please no Americans unless you lived in UK), if I accidentally bumped a car while leaving a parking space and drove off and the other party

The other car my not have been insured the driver may not have had a licence maybe it wasn't his car, it may have had false plates, did you take the other cars details I don't think so.

If I was in your position I would be worried that there's something wrong it doesn't add up

Unless someone is hurt, it's entirely a civil matter over property, so the police will not be involved. The delay for hearing from the other party's insurance company could be fr any number of reasons: no claim has been made; the other party has gone on holiday; they are not insured; the insurance company is just slow. If the repair is expensive, you will hear soon enough.

How obvious is the damage to the other car? Modern materials on a steel frame plastic etc, give and sometimes spring out despite the steel panel that they are mounted to being damaged. That internal damage often can't be seen.

Filing a claim for minor damage may fix the problem, but your insurance may go up and you end up paying more to claim it.
Try not to do that, it is not nice at all. If you hit someone own up to it.

Depends if the other party wants to claim on your insurance I suppose