I fucking hate bullies and the weak and pathetic attitude of schools in dealing with it. What makes it worse is that you've raised it with the school and all they've done is pay you lip service. I really hope this comes back and bites them on the arse.
The hospital however have played a blinder because the school have a duty of care to your son whilst he is on educational premises and will be shitting it due to their apparent lack of action, which resulted in the flagging up of a safeguarding issue. A bust lip is one thing, but for him to have suffered a broken arm, which incidentally in criminal law would be a Section 20 Assault (GBH) for us older folk, does not reflect well on their standards and their ability to protect victims of bullying. Even though you seem to be reasonable don't be with the school, they'll try andxworm their way out of it, but they've let your son down and need their arses kicking. A mention of media, local MP or the borough education office always seems to assist in making sure the Head teacher gets off their arse and deals with the matter, as opposed to delegating it to a head of year or pastoral care. Good luck and good luck to your lad, does he not fancy taking up boxing or a martial art, that's the best way for him to distribute some future justice.