I'm assuming the child was in a relatively well condition seeing as the mother accepted the decision to take her home and bring her back the next morning, if the mom had serious concerns about her daughter then I imagine she would have gone to A&E.
I think the problem is that people, naturally, trust GPs unless the GP betrays that trust. If a GP says they will see a child tomorrow, wouldn't most people accept it? Not all parents have the character to stand up to a GP receptionist and say they are staying there until they are seen.
It's also not easy to make that decision whether or not you need an ambulance - it is easy with hindsight to say you should have called an ambulance (been there, done that) but in the midst of the situation when it's your own child it's nowhere near as clear cut because you are trying to care for them, trying to deal with your own emotions and I think you are likely to accept what the GP says unless you are as stubborn as I am.
The GP had a duty of care and she failed to fulfil the obligations of that duty of care.