I'd want some more details before making a proper judgement and possibly before going in all guns blazing. As mum, first instinct is to be angry, and it's not always to be dismissed as "overprotective or overreacting." It may well be that it has been seriously mismanaged. But it could also be that it was managed appropriately, and you are (understandably) actually more upset that he has been hurt.
Firstly, how much time has passed between the accident happening and you being called?
Second, in what way do you say it was an obvious break? Was their an obvious misalignment? Was bone protruding through the skin?
When you say it was manipulated... how? As in, they put it into a sling? They asked if he could move it/fingers/wrist? Did they put ice pack/frozen peas on it?
What were the circumstances of the accident? Was it during pe, for example or whilst running around on the playground?
Did they call an ambulance, If it was "very obvious" that he needed to go to the hospital, These things are assessed, and if an ambulance is not deemed necessary they would be told as such (literally, "this is not a life threatening emergency and an emergency ambulance will not be sent, make your own way.") Did this happen, then they rang you? Was staff preparing to take him to a and e instead of an ambulance if they had been told no, or there was a long wait time for it?
With regards to accident forms (which will almost certainly have been done, even if they didn't give you a copy), If it had just happened, they may have been more concerned with dealing with him than filling out the paperwork as you would expect.) But has this been reported to RIDDOR?
Also, ask yourself, in that moment, what would you have done differently if he'd had the same accident/injury at home or at the park? How do you feel it should've been dealt with? If it was to happen again, what would you want doing differently?
I work for the ambulance service, so obviously, I have dealt with things like this ranging from minor to very, very serious, just for context. Also used to be deputy head teacher.