Your daughter has a UK passport and an Italian ID Card.
- She can enter the UK with the UK passport.
- She cannot enter the UK with the Italian ID card.
- (There's no exit passport control in the UK)
- She can enter and exit Italy (the whole EU+Schengen Area, really) with just the Italian ID card.
- She could really also enter and exit Italy with the UK passport, though, but that may trigger a number of issues, so we'll just set this possibility aside.
This is the way to go:
What Brexit changed is that she cannot enter the UK with an Italian ID card anymore (she could before Brexit). It does not change in any way the fact that she can use her Italian ID Card to enter or exit Italy.
- Airlines will not allow her to board a flight to the UK if she presents only her Italian ID Card to them, that's why she should show them the UK passport in that direction (they won't care about the Italian ID Card).
- Italian passport control really only care about the Italian ID Card, but to be helpful, they may tell her she cannot go to the UK with just that ID Card (because every single day they see people trying to fly to the UK with just an ID card, and being turned back at the gate).
It has no bearing whatsoever on letting her in or out of Italy. And just showing the UK passport (or most probably even just telling them she has one) will be enough to assuage their fears.
Note however that there may be restrictions for minors travelling alone on either side, which I'm not familiar with. In some countries like France, a passport used to count as ID + authorization to travel abroad while an ID Card required a separate authorization. I have no idea what the situation may be in Italy or the UK in that respect.