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

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.