I'm very sorry to hear about your daughters injury, I hope she has a speedy recovery and is back to training soon! I coach gymnastics so wanted to share my view and suggest what I would do in your shoes.
Protocol differs from club to club but I'd expect that your club would have an accidents / injury policy, first aid trained staff (though not necessarily all sports first aid qualified - some may just have emergency first aid which doesn't cover fractures etc in any detail), an accident / injury logbook and a welfare or child protection officer. Details of the accident policy and the welfare officer's contact details should be in your parent information pack or on your clubs website so you should hopefully already have access to this information.
I'd double check facts with your daughter - when she thinks this bad landing/fall happened, what was she doing at the time, where was she in the gym, what did she tell the staff, what did they say/do etc - then contact the head coach / club director to explain what your daughter has told you, what you experienced when you picked her up and what happened at the hospital and to request a meeting to discuss. I'd expect the head coach to then speak to the staff to get their account of what happened so you can discuss, along with the accident report, at your meeting. This will enable a much better conversation - if the club can't produce an accident report or provide any explanation lf what happened then you make it clear that this isn't acceptable to you and you find another club for your daughter to train at. If the coaches account is different to your daughters - e.g. fall happened 15 mins before end of session, no obvious signs of injury and your daughter kept saying she was fine to rest and see how she felt until you arrived (though this doesn't excuse the not very detailed handover) - then you discuss this with your daughter, see what she says and take it from there. It does seem strange to me that a 10 year old would sit out for 2 hours without asking to go home - would your daughter be nervous or scared to ask the coaches if they could contact you to collect her?
Gather as much information as you can before you decide how to proceed. The hardest thing about being the parent (and the head coach / director / welfare officer) in this situation is that you weren't there to witness the incident yourself so you don't know for sure what has happened. Accidents often happen very quickly in gymnastics so even those there at the time may not be able to give you an exact explanation of what happened.
The comments above re "what if it was a head injury" "you should seek legal action" are not helpful in my view - it is very unfortunate that the coaches didn't deal with your daughter's injury appropriately but that does not automatically mean that they'd fail to help an unconscious child with a serious head or neck injury. Think about your overall impression and experience of the club - have you had any cause for concern in the past? How have other injuries been dealt with? A couple of years ago we had a child trip and fall on the way back from the bathroom, she bounced back up, we asked her if she was ok, she said yes and finished the session - woke up the next morning with a very sore wrist which turned out to be broken. We all felt terrible that we didn't pick up on this but it doesn't mean that we are negligent or unsafe.
Think about what you expect from speaking to the club / complaining. In my view your priority should be establishing whether you continue to trust the club with your child's safety - it may be that this injury leads them to provide additional staff training or amend their accident policy to aim to ensure this doesn't happen again with anyone else or it may be that your daughter could attend a different session with different coaches if you don't trust the particular individuals that coach your daughter's class. Or if the trust is broken then you find another club for your daughter to attend so that she can continue to take part in an activity that she loves.
Hope this helps.