Your contract says that you must work extra hours on demand for no money? That clause doesn't have much hope of standing up in court.
I think you'd have a good shot at fighting your corner if you spent your evenings drunkenly shouting at the TV. That you spend them caring for someone else strengthens your argument enormously.
Employment law is strongly biased in favour of the employee. Disciplinary action would be interesting - take a union rep along if it goes that far. Otherwise the stance of "no, I'm not changing my contract to suit you, as I must care for my mother" seems solid.
edit: Turns out I'm badly wrong there. The contract term is fine. Look into "associative discrimination" claims, under the equality act 2010. Your employer is required to make "reasonable adjustments", though they may disagree on what constitutes "reasonable". The other option is "implied breach of contract" claim e.g. unfair dismissal if you walk out because of the change. Working the same hours for 5 years is an implied term of the contract - though again they may disagree.
Maintain a written record of everything your employer is telling you and ask for a copy of the firms disciplinary policy (if you're lucky, they haven't written one). Ensure that they are then acting in accordance with it.
ACAS "should" be useful for free advice, otherwise there's the citizens advice bureau or someone from your union (if you're a member). Proper, non-computing forum legal advice would seem prudent at this point.