What kind of crappy parent would NOT take their own children to the Doc/A&E but insist the other parent - who doesn't even have them with him at the time - does.
If she were really worried about the children, to the point that she felt ONLY HOSPITAL was the answer, she should take them. no question at all.
She has in the past asked this man to have his kids on weekends that weren't 'his' and when he's said that he can't , the kids have rung in tears because she has told them he doesn't care about them and that they would miss things that are important to them because he can't dance to her tune. She tells him that he doens't care about the kids if he has other plans.
OP - you have the kids with you at the moment? tell them how much you love them and that if you can ever do x y or z that you will, but that sometimes there are plans made that are set. Their mother is trying to create trouble for whatever reason, and that sometimes she says things that are not entirely true but they need to trust you to know that you will always do what you can for them. Start a dialogue of truth with them.
Then the next time you get this ridiculous woman making her problems YOUR problems, or trying to create drama, let her jog herself on. Say no and mean no. If she tries the hospital thing again, tell her to call 101 for advice.
My OH Ex was a little like this, threats of all kinds of things, that she'd stop holidays, or visits etc, OH told her OK, no problem, he'd go on holiday anyway, and carry on with his weekends as he'd planned.
Grey Man - the SAS use this as a technique - make yourself as dull as poss, don't react, don't stand out, give her NOTHING in the way of drama or response
she is jealous or bitter or both. that's her issue.