To ask if you'd accept this?

[pinklite]

A couple of months ago I had car accident which wasn't my fault. My car was written off and I suffered with whiplash and back pain as a result of the accident and am still struggling very badly with anxiety. I have considered seeing my GP as I'm constantly having irrational thoughts about bad things happening to myself and my family, as the accident made me realise that I'm ultimately not in control of other people's actions which could cause harm. My sleep has also been affected and even though it's been a few weeks, I still replay what happened in my head almost every single day. Not to mention I'm now really anxious driving anywhere.

My solicitors have been in touch and have put forward a quote of £1.5k. Is this about normal for incidents like this or should I expect more?

Reading that, the tariff seems to apply to injuries of up to 24 months duration, and if your injuries are longer lasting than that you will have to be prepared to go to court and prove it.
In my case, which predated the above legislation, I had to wait until 3 years had passed to "prove" that my injuries were permanent. My settlement, which was substantial, was to cover future therapy for physical/mental injuries and loss of earnings/pension rights - I had to give up my job. Not sure how much was in that settlement for pain and suffering, as that part of the claim is intangible.
In short, if you think you are entitled to more, you will have to prove it and it will take time.

It's worth going through your union to represent you. They'll not take anything, any solicitor will take upwards of 30%.