As you probably know, you generally get one go at a flight change, and once that's used up BA will routinely charge for further changes.

I'm not suggesting you shouldn't call them to ask them to listen to the call to see what your mother was told, but I think you should equally be prepared to be disappointed.

That said, BA would struggle to argue the F to J downgrade was voluntary, so you should be able to claim for a 75% reimbursement under UK261. You are likely to need a legal route to get this, though. Unfortunately any threat of this won't be something that influences the agent you speak to - quite the opposite in fact, all it'll do is cause an immediate withdrawal of any CS resolution and a referral to legal - so you should only go down that route once you're satisfied there is no other option.

I suppose CEDR might be another option, but likely your mother will have flown by then anyway given the backlog of cases.