I'm an elementary school teacher. At my district, a physical note is sent home with the child anytime the child goes to the nurse's office, but who knows if the child ever gives that note to the parent. Someone here said their nurse's office sends an email for every single visit, but there is no way our school could do that given the volume of students visiting one nurse per day. Suspected injuries of any kind, especially head bumps of any kind, warrant a phone call home from the nurse. It's then the parent's call whether to pick the kid up for immediate x-rays. That should have happened for your child, but didn't.
I agree with the almost unanimous opinion here that non-injury nosebleeds are not something to call home about. Kids get them all the time and ordinary ones are not dangerous in any way.
It seems you're still angry about the broken arm not being reported, which is understandable. The ideal time to (calmly) complain was when it happened. No yelling or profanities necessary- the nurse did not break your child's arm, remember. The approach to take, even now, would be to discuss your disappointment and concern with the situation, and the fact that it was serious enough to still bother you today.
Your child's school district clearly has a personnel issue, in that there is only 1 nurse for 3 schools (and the fact that you're not allowed in is weird, too). Every public school should have a full-time RN on duty. I would personally follow up with both school and district administration about your concern about the inadequate staffing which undoubtedly contributed to this incident. It could be the district's choice to allocate staffing this way: more overpaid pencil pushers in central office, and fewer nurses on the ground in schools. If parents don't complain, they assume parents are satisfied with the status quo.