All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
If the father gives money to his young child as a gift and he saves this money for him in a bank account, and especially if the account is in the child's name and the father declares that this money belongs to his child, then the gift has been concluded, and the father's possession of the money on behalf of the child is sufficient. Ibn Qudaamah said: “If the father gives his son a gift, he acts on his son's behalf to receive and accept it if needed.”
If the child's money has reached the Nisaab (the minimum amount liable for Zakah), his guardian is obligated to pay Zakah on it. This is the view of the majority of the scholars from the Maaliki, Shaafi‘i and Hanbali Schools of jurisprudence. Their evidence is a report in At-Tirmithi that the Prophet, sallallaahu ‘alayhi wa sallam, said in an address to the people: “If any of you is the guardian of an orphan who has money, he should trade with this money and he should not leave it to be consumed by Zakah.”
When the child reaches puberty, he will either have access to and control of the account, in which case he is obligated to pay Zakah by himself if his father had not paid it on his behalf with his consent, or he will still not have control of the account due to his parents blocking it; in this case, if they did not pay the Zakah (on his behalf), he must pay the Zakah for the previous years when he takes possession of the money.
Allaah Knows best.