There is no specific definition for a "reasonable amount of time" but if it has been more than a few days or over a week you should continue to contact the at-fault party's claims adjuster that is working your claim to request information on your claim.

If you are need to get a rental car on your own while your car is unable to be used then make sure to inform the other party's insurance company and advise them you are expecting to be reimbursed. The insurance adjuster should be able to tell you the amount per day (rental car reimbursement amount) that the person's policy limits will allow for.

According to the Georgia Insurance Commissioner, in third party situations, a claimant may be entitled to car rental from the date of the accident until the car is, with dispatch, repaired. Furthermore they go on to say that a recently enacted statute states that any insurer, upon acceptance of liability, pursuant to any automobile liability or motor vehicle liability insurance policy, shall pay reasonable benefits for losses, including total losses, to a third party for loss of use of such motor vehicle.

So if the other party's insurance company has accepted liability for the damages their insurer caused to your vehicle then a rental car should be provided during the loss of use of your auto. For more information on this topic as well as the length of time the at-fault's insurance company has to contact you with information on the repairs of your car, etc get in touch with the GA Insurance Commissioner.