I don’t think the school or after school club are breaking any laws, though. They probably haven’t used any force. Even false imprisonment, as a crime, would need them to have criminal intent (which is different from just being wrong about whether it’s okay for her to walk home or not) and that would be hard to show. I think it’s highly unlikely there is anything illegal in what the school is doing.
OP isn’t endangering her child by wanting her to walk home alone, so social services aren’t going to be interested in pursuing her if the school calls and neither are the police.
It’s a bit of stand off. Neither of them are breaking the law, so this becomes a civil dispute between the OP/her child and the school and after school club.
The school have no authority to require OP to pick up her child and they can’t incur charges on her behalf. OP might be able to seek a court order requiring the school to release her child at the end of the day but that costs money. Given one of the reasons she no longer wants to use the ASC is the increased cost, that’s unlikely to be a useful solution for her.
Probably her best bet is to refuse to pay the after school club charge (and make sure they know, in writing, that she refuses the after school club service and won’t be paying for it) and just collect her child when it’s convenient, saying each time that she should not have been sent to ASC and OP will refuse to pay.
An alternative is to play a game of brinkmanship, refuse to collect her from ASC and just keep reiterating to ASC, and SS if they are called, that her child is perfectly capable of walking home and shouldn’t be being kept at school. The big concern there is the impact on the child. The school clearly don’t put that aspect above adherence to their policy, but OP is probably more concerned about her.