Whether zakāt is necessary on money saved for children depends on the intention when setting the money aside for them. If the money has been given to them as a gift, such that they are the owners, then zakāt is not necessary as they are not adults. This does not change even if the money is in an account which they do not have access to.[1]

On the other hand, if the money has been put aside for them to cover their future costs without them owning the money, then zakāt will be necessary on whoever has set aside that money.

And Allah knows best

Ibrāhīm ibn Muḥammad

[1] قال التمرتاشي: وهبة من له ولاية على الطفل في الجملة تتم بالعقد (تنوير الأبصار: 5/694)
قال ابن عابدين: قال في التتارخانية: فلو أرسل العبد في حاجة أو كان آبقا في دار الإسلام فوهبه من ابنه صحت فلو لم يرجع العبد حتى مات الأب لا يصير ميراثا عن الأب (رد المحتار: 5/694)