There are a lot of answers to this question. I can only speak to auto claims. If you’re filing a simple auto damage claim with your own company, it should be possible to start repairs within a few days.
Third party claims, which are filed with an at-fault party’s insurance company, can take a little longer. Even a simple third party claim requires some additional investigation into liability, which can take from a few days to a few weeks. Do you want somebody to make a claim against your insurance and pay before they confirm you did it?.
Adding a medical component can add some time, as well. Just like damage claims, there are first party coverages (Personal Injury Protection, or Medical Payments, depending on state), and third party (Bodily Injury Liability). First party claims can often receive and process bills as they go, usually in a couple weeks to a month. Conclusion of the entire claim is more related to medical treatment duration than anything else.
Third party bodily injury claims are a little different. Any settlement is a lump sum payment after treatment is over, so not much happens while you treat, other than the insurance adjuster calling for updates and ordering bills and records, if you authorize him to do so. The level of cooperation by injured parties, the presence of an attorney, the adjuster’s workload, the medical provider’s mail response time, and a whole host of other things can affect the time frame by a large margin.
Basically, an insurance claim can take a very short or very long time. If you’re persistent, and/or you’re working with a good company, you should at least be able to get a good idea of what the time frame is expected to be and why.