How long do you have after a car accident to report and file a claim with the at fault's persons insurance company with no police report?

With any claim, you will want to report the incident ASAP. When I've had incidents, I've always done it same day. Anyhow, your specific policy will let you know how long you have. and it's usually 30 to 60 days depending on the carrier. At any rate, you will want to contact your own insurance first and they should then contact the at-fault's carrier, unless you lack insurance which is completely dumb in the first place. The reason you will want to report ASAP is because most insurance companies require that you "keep the car from further incidents and damage" until the claim is settled and/or filed. The reason for this is to prevent fraud.

I think the first paragraph is explaining how quickly you should tell your own insurer. The question, however, seems to suggest that the person At Fault is another car driver and as such, in England and Wales, you have to commence a claim at court for personal injury within 3 years of the date of the accident.

Please see the link below for the process of making a claim which will include contacting the driver at fault and the motor insurers.

The standard Wording in a personal auto policy offered in the US has no stated time period for the reporting of a claim. It will require that you are "reasonably prompt."

What reasonably prompt means varies by jurisdiction and circumstance so reporting to your insurance company ASAP is the safest thing to do.

Your policy will have a section that outlines your duties after an accident or loss. Failure to perform those duties can result in an automatic denial of your claim. So it is important to file promptly and to do the other things outlined in that section of your policy.