1. As others note, it is extraordinarily unlikely that anybody but a UA agent and then only a UA who actually looks, would see the note. It is even more unlikely that any other carrier would honor the request.

2. As to your original question. If you had booked on one itinerary, the baggage allowance for the originating carrier of the first segment (UA) would have carried through for the entire remainder of the itinerary.

Bottom line is that mom ought to be prepared to pay on the return.