I don't think they can make decisions like that any more than I could - they have no power invested them to compel anything.
They aren't judges. Even Social Services, which is part of the civil service, and not a private enterprise (like mediation companies are) can't do this - they'd have to apply to a judge first.
This woman has utterly failed today to mediate between you and Toad, which is her job, and that makes me quite glad she doesn't have any powers to be honest!
Besides, how on earth is she going to hand over a nearly 15 year old -she'd have to bind him up before doing that, which is technically assault!
Your mind is on overdrive (and I know I do the same thing - been doing it all day to be honest!) worrying about what might happen. I think pulling out of mediation is necessary. No one is going to take your children away from you unless you are doing something very, very wrong. You aren't - we know that, you know that.
I'm sorry you are feeling poorly - get yourself into bed if you can, painkillers and rest. You've had a horrible day, and I imagine sleep is a long way from coming to you, but this will end.
It's always darkest before dawn as they say.