If a child is in your care and suddenly and unexpectedly becomes ill or injured, it's the job of you, until the parents arrive. Whether you are teacher, babysitter, scout leader or swimming teacher.
This.
Yes the parents should come asap, particularly if it is on the more serious end of illness or injury.
But the responsible adult must provide the necessary care in the interim - as Scout Leaders we have trained in first aid for this purpose (we may be up a mountain so not necessarily easy for parents to get to their DC!), and I have also trained and act as an occupational first aider for work, where the organization has a duty of care to its employees.
It is not ideal to take away from teaching time - but if a DC falls over in yard and breaks a bone in their leg (as happened in our school last week - my DD was telling me the gory details!), then they need to be cared for until either the parent arrives or medical help arrives or the child is brought by school staff to seek medical help.
Some schools have some more resources than others to juggle in such instances - using resource time for a child to do a "whole class" aspect of the DCs programme, splitting a class between other teachers, using support staff to care for the ill child, principal stepping in somewhere and ignoring other paperwork/admin issues until later....just as a few ideas immediately springing to mind. You don't expect such a situation to arise on a daily basis, but you should expect to have bumps, knocks, cuts, sore tummies, sore heads, D&V, etc on an almost daily basis and have some sort of procedures worked out for dealing with the run of the mill events, and a second procedure to kick in when something more major happens.