Answer
Wa`alaykum as-salamu wa rahmatullahi wa barakatuh.
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
In this fatwa:
1- Zakah has nothing to do with the age of the owner of the property.
2- Once the property reaches the nisab (Zakah-payable amount), Zakah has to be paid.
3- The father or the guardian of the child should pay it on his behalf.
In his response to your question, Dr. Muzammil H. Siddiqi, former President of the Islamic Society of North America, states:
Zakah on Children’s Wealth
Zakah is obligatory on the nisab whether the owner is an adult or a minor. If a minor’s money reaches the nisab of Zakah, it will be the duty of his/her guardian to pay Zakah for it.
If parents have put aside some money for their children or have given them some form of wealth, it remains the parent’s money. As such, they may use it in case of emergency or for any form of necessity.
It is reported in a hadith that a young man came to the Prophet (peace and blessings be upon him) and said, “My father has taken my money.” The Prophet (peace and blessings be upon him) said, “You and your money belong to your father.” (Ibn Majah)
The meaning is that parents do have a right to utilize the money of their children, especially if it is they who gave it to them.
Allah Almighty knows best.
Excerpted, with slight modifications, from: www.pakistanlink.com/religion