Is it obligatory to break the fast with something from the Udhiyah on the Day of Sacrifice? And is it permissible to break the fast with the mother's Udhiyah if the slaughter of one's own Udhiyah is delayed?

It is Sunnah for one who offers a sacrifice (Uḍḥiyah) to delay breaking their fast on the Day of Sacrifice until they can break it with something from the sacrifice. It was narrated from Buraydah (may Allah be pleased with him) that the Prophet (peace be upon him) "would not eat on the Day of Sacrifice until he had slaughtered." This delay is recommended, not obligatory. Therefore, there is no harm in eating from the sacrifice of one's mother or siblings, or anything else.