You should always ask for a written acknowledgement that they have received your documents. For example, never return your tax disc to DVLA before your car insurance company have given you a written acknowledgement. By returning the tax disc to DVLA you can get a part of the tax reimbursed.
Sometimes the car insurance company engineers can take a very long time to have your damaged car examined. So don't dispose of your car before the insurance company says it's okay. Don't forget to have it in writing from them. Usually the claim handling is straight forward.
When it takes longer time to resolve
Sometimes the circumstances of the car accident could be pretty complicated. If for example your car was driven by another family member. Perhaps your car insurance policy cover this or your family member have to use he or his own insurance cover. The problem now is that you will have to make a claim to the other car insurance company. The result could be that your family member did not cause any financial loss and therefore will not your family member's car insurance company do anything. Your own car insurance company will perhaps not do nothing at all because they cover you only. So the result end up with it is you who incurred financial loss. With other words, you have to make a claim to your family member insurance company which could end up in court. This sort of procedure takes time because you have to wait for the courts final decision.
AA car valuation is a hard issue. The car insurance company will of course try to minimise their outley and the simple reason is of course that your car insurance company want to make a good profit. The car insurance valuers looks into their value books. The amount is also many times lower than what it might cost you to replace it by your own. You can always contact a reputable garage for a independent valuation of your car. If it turns out that your car insurance company engineer valuation is lower than the independent engineer from the reputable garage. You should consider response and challenge the insurance company's valuation of your car.