State laws can differ on what is required of an insurance company during the claim process. Some states have a set time limit in which insurers must get things accomplished when working claims. In other states there is no time period set.
If a state does not have a set amount of time in which they must look at the car or at least accept liability and then settlement the claim or have repairs started than most at least say that the insurance company must do these items within a reasonable amount of time.
If you are using your own insurance company than the terms of your policy may state how long the insurance company can take to view the vehicle and take care of your claim.
To find how long an insurance company has in your state to handle your claim, contact the insurance regulatory body. If there is no set time, see if they can offer you consumer help on how to get the at-fault party's insurance provider to move along at a swifter pace so that you may either get your vehicle repaired or be offered a settlement. You may also want to call the insurance adjuster assigned to your claim and see what the hold up is since 3 weeks is a bit long.
So while some states may only require insurance companies to have the car looked at and repaired (or if totaled out to offer a settlement) within a "reasonable" amount of times, some states give a specific amount of time.
For example, in Louisiana state laws requires that, unless there is a catastrophe, the insurer must initiate loss adjustment within 14 days after that company has been notified of the loss by the claimant.
Washington state laws say that every insurer shall complete the investigation of a claim within thirty days after notification of claim, unless such investigation cannot reasonably be completed within such time. All persons involved in the investigation of a claim shall provide reasonable assistance to the insurer in order to facilitate compliance with this provision.
North Dakota's Insurance Department states that there is not a set time provided for the settlement of a liability or physical damage claim, as many factors must be considered when investigating an accident. It is in the best interest of the company to investigate any loss promptly so that valuable evidence is not lost or destroyed and so that incurred costs, such as storage and any loss-of-use obligations do not become excessive.