Sorry about the emergency, plus the hassle of the ticket and travel issue. Hope everyone is ok. Couple of things: BA has always been extremely accommodating when I’ve had medical issues. Admittedly these were always my own issues. But I think they are also flexible when it is immediate family: children, spouse, etc. The other major thing is that a full fare J ticket should be extremely flexible and cancelable without fee. Seems nuts that they’ve apparently told you it has essentially zero flexibility.