My understanding is that Cycling UK allows you to join and then use their legal services to pursue an already-occurred collision:
https://www.cyclinguk.org/
And, as others have put, expect it to take some time. The police will take 3-4 weeks before they get the paperwork done, as they already have the driver's details. It could then take 12-18 months for a conclusion should it go to court, although I'd suggest from what you've put elsewhere it won't. It's more likely to be 'no further action' or the driver being offered a Driver Improvement course.
In terms of a claim you need a written assessment from a bike shop of any damage. That's needed for an insurance claim.
You also need to be seen by a doctor for any personal injury / or having photographed these. It'll take a few days for bruises to show but if it's more than bruising you should get a medical assessment.
For my rear-ending, where a Range Rover mounted the pavement to bypass traffic and took me out in the process, I chose simply to cover bike damage through the LBS (new rear wheel, new bar tape). I've had a negative experience with British Cycling's legal team in Leigh Day and wasn't injured. Even that took 2-3 weeks to resolve.