What's wrong with directly stating "anticipating their need?" Or, "anticipated his need", to better match the example? "My mother anticipated that my father's battery would need to be charged." This captures the intended meaning exactly. I don't see a need to rely on just one word.
"Foresaw" works pretty well, but it's not an adjective. "My mother foresaw that my father's battery would need to be charged." My thoughts are that "foresightful" doesn't work, because "foresightful" pulls in too much of a sense of "having foresight", which can mean literally being able to see the future in a supernatural sense, and that doesn't work for a case of reasoned foresight.
To be descriptive, a phrase might be better than trying to find an adjective. For example, "My mother very often anticipates other's needs." Or, "My mother is very good at foreseeing what others will need."