Hi, I developed fibro after an accident, although not a car accident. I also sued for compensation.
When my doctor suspected fibro I started meds (amitryptaline and naproxen), and was referred to rhuemotolgist. The rhuemotolgist literally did a simple physical exam and pressure point test, and asked a lot of questions (blood tests and scans had ruled other things out). He increased my dose of amitryptaline, gave me leaflet and referred me to specialists physio. That's it.
I had the option of going private (and potentially be able to claim it back) but decided against it as I didn't want to end up having to pay for any long term private treatment (like the physio £40+ per session) and didn't want to have to switch to nhs half way through treatment if things got too expensive. Also if they wanted to do their own or additional scans it can get very costly (I had a private arthogram scan once which was billed at £500)
In your position I probably would try and see if GP will increase your medication dosage whilst you wait for the referral to come though. And ask about other treatments like hydrotherapy.
In terms of the insurance claim it is virtually impossible to prove the cause of fibromyalgia. An independent GP has to say on the balance of probability's the accident was the cause - and generally they won't as there are too many factors, and other potential causes. Also their is thinking that people can be pre-disposed to conditions like fibro, so they said the fact I had IBS, hyper mobility and had suffered with stress previously were all indicators, and therefore fibro is not the accidents fault. They will have access to everything in your medical history to use against you - the fact I had neck pain for 1 week when I was 9 was bought up.
I took a settlement agreement in the end, as I wasn't going to prove the fibro link, and could end up loosing money if I went to court.
If you can't prove the fibro link you won't be able to claim back the cost of private treatment for fibro in your insurance claim. And any money you receive in compensation may not cover cost of private treatment either.
Sorry to be a bit a doom and gloom! Just be very wary and don't take on any costs that you can't comfortably cover if things don't go your way.