Is my mother sinful and blameworthy, and is she considered the cause of my aunt's death after she made her swear on the Qur'an regarding the theft of her gold, and my aunt later suffered from cancer and epilepsy?

The fundamental principle is the innocence of a Muslim, so it is not permissible to accuse them except with valid evidence. If the mother accused your uncle's wife without valid evidence, she is sinful and must repent and seek forgiveness for that woman. This accusation does not necessarily have to be the cause of her death, and the fundamental principle is the presumption of innocence, so there is no sin upon the mother regarding her death.