A small child follows the religion of whichever of his parents is Muslim
Question

                I was wondering whether it was necessary for a child raised by two muslim parents to declare their shahada upon reaching puberty. What about a child raised in a home where the father is muslim but the mother is not and the father has not enforced the prayer from the age of 10 and the children do not pray?.
Praise be to Allah.Firstly: The 
    child who is born to two Muslim parents is ruled to be a Muslim, according 
    to scholarly consensus. 
If the 
    parents have different religions, then the child follows the one who is 
    Muslim, whether it is the father or the mother.  
Shaykh 
    al-Islam Ibn Taymiyah (may Allah have mercy on him) said: If the child's 
    parents are both Muslims, then he is Muslim too, following his parents, 
    according to the consensus of the Muslims. The same applies if his mother is 
    Muslim, according to the majority of scholars such as Abu Haneefah, 
    al-Shaafa’i and Ahmad. End quote from Majmoo’ al-Fataawa, 10/437. 
It says in
    al-Mawsoo’ah al-Fiqhiyyah al-Kuwaitiyyah (4/270): The fuqaha’ are 
    unanimously agreed that if the father becomes Muslim and he has young 
    children, then these children are to be regarded as Muslim, following their 
    father.  
The majority 
    (the Hanafis, Shaafa’is and Hanbalis) are of the view that what counts is 
    the Islam of one of the parents, whether it is the father or mother, so the 
    children are to be regarded as Muslims, following the parent, because Islam 
    should prevail and not be prevailed over, because it is the religion of 
    Allah that He is pleased with for His slaves. 
Secondly: 
    When the Muslim child reaches the age of puberty, he is not required to 
    utter the Shahaadatayn again.  
Shaykh 
    al-Islam Ibn Taymiyah said: The Muslims are unanimously agreed that if a 
    child reaches the age of puberty as a Muslim, he is not required to renew 
    the Shahaadatayn. End quote from Dar’ al-Ta’aarud, 4/107. 
And he said: 
    The early generation and the imams are unanimously agreed that the first 
    thing to be enjoined upon people is the Shahaadatayn, and they are agreed 
    that if a person did that before reaching puberty, he should not be enjoined 
    to renew that when he reaches puberty. End quote from Dar’ al-Ta’aarud, 
    4/107 
But if after 
    reaching puberty he says or does something that indicates that he is not 
    content with Islam, then he is to be regarded as an apostate and is to be 
    treated as one who has apostatised from the religion of Islam. 
Shaykh 
    al-Islam said: In terms of worldly rulings, the child comes under the same 
    rulings as his parents, because he is not independent. When he reaches 
    puberty and speaks words of Islam or disbelief, then he is to be judged on 
    that basis, according to the consensus of the Muslims. If his parents are 
    Jews or Christians, but he becomes Muslim, then he is a Muslim according to 
    Muslim consensus. If they are Muslims and he becomes a kaafir, then he is a 
    kaafir according to Muslim consensus. End quote from al-Fataawa al-Kubra, 
    1/64 
Thirdly: 
    When the child reaches the age of seven, his parents should instruct him to 
    pray and encourage him to do so, because of the report narrated by 
    ‘Abd-Allaah ibn ‘Amr ibn al-‘Aas (may Allah be pleased with him), according 
    to which the Messenger of Allah (blessings and peace of Allah be upon him) 
    said: “Instruct your children to pray when they are seven years old and 
    smack them if they do not do it when they are ten.” Narrated by Abu Dawood 
    (495); classed as saheeh by al-Albaani in Saheeh Abi Dawood (466). 
Al-Nawawi 
    said: The imams said: It is obligatory for the fathers and mothers to teach 
    their children about purification, prayer and other laws after the age of 
    seven, and to smack them if they do not do them after the age of ten. End 
    quote from al-Majmoo’, 3/11. 
Ibn Qudaamah 
    said: This discipline is prescribed for the child in order to accustom him 
    to prayer, so that he will feel comfortable with it and get used to it, and 
    he will not neglect it when he reaches puberty, but it is not obligatory 
    upon him. Al-Mughni, 1/682 
If a child 
    does not pray before the age of puberty, that does not put him beyond the 
    pale of Islam, because he is not accountable for doing it and it is not 
    obligatory for him. 
Shaykh 
    al-Islam said: Prayer is not obligatory for a child, even if he has reached 
    the age of ten. This is the view of the majority of scholars. 

    Al-Ikhtiyaaraat al-Fiqhiyyah, 1/32; see also the 
    answer to question number 1994. 
Based on this, the child who has a Muslim father and a non-Muslim mother is a Muslim. If he reaches the age of ten and does not pray, he is not a kaafir because of his not praying, because he is not accountable for that until he reaches the age of puberty. If he reaches the age of puberty and persists in not praying, then he is an apostate from Islam because of not praying.  
And Allah knows best.