Is the guardian obligated to offer expiation for not paying the zakat on an orphan's money that reached the minimum threshold (nisab) and a full year (hawl) passed over it for years? And who is responsible for paying the overdue zakat? And is the guardian sinful if the orphan neglects to pay it after receiving the money?

Since you are the guardian of your brother's wealth due to his lack of maturity, you should have paid zakat on his wealth if it reached the nisab (threshold) and a hawl (full lunar year) passed on it. Now, you must repent for not paying the zakat on time. If your brother is mature, he is responsible for paying the overdue zakat; otherwise, you must recover his money and pay his zakat. Guardianship ends when the minor reaches maturity, i.e., becomes trustworthy in managing his wealth. This maturity is not determined by a specific age but is demonstrated through testing and experience. If his maturity is proven, his wealth should be handed over to him, as stated in the Almighty's saying: "And test the orphans until they reach the age of marriage. Then if you perceive in them sound judgment, release their property to them." If he reaches maturity but is not of sound judgment, his wealth remains under your guardianship until his maturity is proven, as stated in the Almighty's saying: "And do not give the incompetent your property, which Allah has made a means of sustenance for you."