Paying zakaah from a child’s wealth for past years

Firstly:

Whoever has failed to pay zakaah for past years has to pay it, whether he omitted to do so knowingly or out of ignorance. Shaykh Ibn Baaz (may Allaah have mercy on him) was asked about someone who did not pay zakaah for five years out of ignorance. He replied: 

You have to pay zakaah for all the past years. Your being ignorant does not mean that it is waived for you, because the obligation of zakaah is a well known part of the religion which no Muslim has any excuse for not knowing, and the ruling is not concealed from the Muslims. Zakaah is the third pillar of Islam. So you must hasten to pay zakaah for all the past years, as well as repent to Allaah for delaying it. May Allaah forgive us, you and all the Muslims. And Allaah is the Source of strength. 

End quote from Majmoo’ Fataawa al-Shaykh Ibn Baaz, 14/239 

See also question no. 21715 

An exception from that is zakaah on kinds of wealth where there is some scholarly dispute as to whether zakaah is due on them, such as jewellery that is used. Whoever does not pay zakaah on it out of ignorance, or because he follows a Shaykh who says that zakaah is not due on it, then he does not have to pay zakaah on it for the past, but he should pay zakaah on it once he knows that it is subject to zakaah. 

This was also stated in a fatwa by Shaykh Ibn Baaz (may Allaah have mercy on him) who said: We also draw attention to the fact that it is obligatory to pay zakaah once you know that it is required on jewellery, but with regard to the past years before knowing that it was required, you do not have to pay zakaah for that, because the rulings of sharee’ah only become binding after one learns of them, and because there is a scholarly difference of opinion on this issue. End quote from Fataawa Islamiyyah, 2/84 

Secondly: 

The majority of scholars are of the view that zakaah is due on the wealth of minors, and that being a minor does not mean that the obligation to pay zakaah is waived. 

Ibn Qudaamah (may Allaah have mercy on him) said: Zakaah must be paid on the wealth of minors and the insane. That was narrated from ‘Umar, ‘Ali, Ibn ‘Umar, ‘Aa’ishah, al-Hasan ibn ‘Ali and Jaabir (may Allaah be pleased with them all). It was also the view of Maalik, al-Shaafa’i and Ahmad. End quote from al-Mughni, 2/256. 

This has already been discussed in the answer to question no 75307. 

Thirdly: 

Not all shares are subject to zakaah. Some of them are subject to zakaah and in other cases, zakaah is only due on the profits, if they reach the nisaab and one year has passed. For more information on that, please see the answer to question no 69912 . 

If these shares are subject to zakaah, then he must pay zakaah for the past years.  

If zakaah is due on their profits only, then he must pay zakaah on the profits that reach the nisaab, after one year has passed. 

And Allaah knows best.