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.