If no one else has said this yet (cant be arsed to read the responses tbh!) This has ALWAYS been the case for the circumstances you describe, for like the last 20 years minimum....
.... I know this because I worked for the police from 2006-2017 and told people this on a daily basis. I also had a massive car accident in 2004 and was told this.
The technical bit: IF you have a car accident you are not required to report that to the police unless there are injuries involved or the road is blocked. Otherwise you can exchange insurance details and be on your merry way.
Sometimes when people get home and phone their insurance company, their insurance company asks if there were any injuries and the person realises that they will get more money/etc out of their insurance if they say they are injured, so they say their neck or arm or leg hurts, but that there are no visible injuries. The insurance company is on to 'crash for cash' liars and so then demands paperwork from the hospital or a letter from the GP confirning that there is an injury, or at least that one has been reported.
This surprises our poor injured car driver but they decide to go to their GP the next day and fake whine about their sore necks and ask for a letter to send to their insurance company.
The GP is also onto 'crash for cash' offences, and so says well I can write you a letter but I will have to charge you our standard letter writing fee, which they are well within their rights to do. OR they will have a big sign (my GP has it on the glass of the entrance door) saying that if you are coming to see them the day AFTER your car accident (even if yes your injury is genuine) then they WILL charge you their 'private' fee. This is the same as the fee they would charge you to do a private health check for life insurance, or a private health certificate for job interview for specific jobs (like the military, deep sea diving etc), they can also charge private fees for vaccines you don't normally need for the UK too, but need for a border crossing/visa, like Yellow Fever. (Mind it was nearly 20 years ago I got charged for that too so that may not be the same now!!)
And YES in respect of car accidents they do charge it in an effort to weed out people who are NOT seriously injured, or who ARE lying about their injuries for insurance purposes as it often puts off people who are trying to fake their way to a big pay out, in other words the cost/fee puts them off. And yes sadly this negatively affects people who are genuinely injured.
The absolute BEST thing you can do if you are genuinely injured in a car accident is to go to the hospital or GP immediately. Obviously if its a serious injury you may need an ambulance, or one may have been called for you, otherwise make your own way to A&E.
If thats not an option or you dont think your injury is that serious but want it registered, then call your GP and try and get an appointment, if they dont have one ask them if they can register your call and if doing so will mean you wont get a charge if you call for an appointment tomorrow. They may say, yep we'll see you free of charge because you called today and its our fault we couldnt see you. They may say, nope, nothing we can do about it, you'll incur a charge. If thats the case then I would go and sit in a&e until you get seen OR take the charge on the chin, as you need that paperwork to make a claim on your/the other parties insurance.
Most people just arent aware of this because it doesn't happen to most people or you dont have to make use of it! But its totally normal across the country, ALL GPs do it.
And, whats important to remember is that they're perfectly within their rights to do it. They advertised (loudly) that they can and will do it too.
So the absolute best thing to do when youve had a car accident is take stock of your injuries and if something hurts even a tiny bit, then
A ) you need to report it to the police immediately. Police are supposed to attend the scene of any RTC where an injury has occurred.
B ) GO AND GET IT CHECKED OUT AT THE APPROPRIATE VENUE THAT SAME DAY and then you won't get charged private fees like the OP.
Honestly, this is the sort of shit they should be asking you in order to pass a driving test 🙄 😒