If it isnt then it should. As long as you exchange details, then there is no need for the police. What are they really going to do when they attend? They will take an account of what happened and take everyones details. Which is exactly what you can do on your own. If the other party doesnt stop or refuses to give details, then the police do need to attend.
At an accident/incident none of those directly involved will be in the best frame of mind. You may be unwilling to speak to the other party, they may be unwilling to speak to you. Insurance companies will always say don't admit liabilty at the scene.Each party involved can acuse the other of twisting what they said.
The incident was serious enough to warrant the attendance of an ambulance & crew. Why not the police. Unless policing has changed isn't it down to them to decide which if any law(s) were broken. Whom by, and from that work out the guilty party/parties & decide the necessary action to be taken. They also act as impartial statement takers, taking both sides of the story. If the other party doesn't feel like waiting to "swap details" & you didn't have the registration, or witnesses, then its a bit late asking for them to attend.
As I said, I've been there. So what I'm using is what was used on the night, and after, by myself. I'll turn this round at you and ask this. How do you stop someone who has just collided with you, leaving you on the road, not in the best of ways, from doing anything further to you. A driver who when he approaches you tries to get you to put the phone down, slurring his words because he's been on the drink. He's also approaching with something in his hand. Turned out to be a half empty/half full bottle. "Insisting" he doesn't want the police. He will later go on to fail the roadside breath test & as his last line of defence(excuse), before the court hearing was due, claim that he in fact thought you were a bus & was therefore safe to pull out onto?
I'll try to answer one question before you ask. The handset in use that night has an alert key, press that & the call is made. So only the one button to press to get the call made.