Are the marriages of the disbelievers regarded as valid for them and are the children to be attributed to them?
Question

                My understanding is that any children born to parents who have not had nikah are illegitimate. Therefore, that would mean that all non-Muslims living today are illegitimate children. Is that correct?
Praise be to Allah.
    Firstly: 

    Marriages that are regarded by the disbelievers as valid are 
    approved by Islam and deemed to be valid, and the same consequences apply to 
    them as to marriages that are sound according to sharee‘ah, such as 
    legitimacy of children and inheritance between the spouses, and other 
    rulings connected to valid marriage. 

    At the time of the Prophet (blessings and peace of Allah be 
    upon him) many people became Muslim and he did not ask about the details or 
    conditions of their marriages; rather he approved of them, which indicated 
    that they were valid.  

    Ibn Qudaamah (may Allah have mercy on him) said: The 
    marriages of the disbelievers are valid and are approved of if they become 
    Muslim or if they refer to us for judgement, if the woman is one whom it is 
    permissible to marry at the present moment, and the details and manner of 
    the marriage contract are not to be examined and the marriage is not to be 
    examined to see whether it conformed to the conditions for marriage among 
    the Muslims, such as the presence of the wali (woman’s guardian) and 
    witnesses, and the format of the proposal and acceptance, and so on. There 
    is no difference of opinion concerning this among the Muslims. Ibn ‘Abd 
    al-Barr said: The scholars are unanimously agreed that if two spouses become 
    Muslim together at the same time, they remain married according to their 
    original marriage, so long as they are not too closely related by blood or 
    through breastfeeding. People became Muslim at the time of the Messenger of 
    Allah (blessings and peace of Allah be upon him) and their wives also became 
    Muslim, and their marriages were approved of; the Messenger of Allah 
    (blessings and peace of Allah be upon him) did not ask them about the 
    conditions of their marriage or how it was done. This is something that is 
    known through mutawaatir reports and is something that is well known to all 
    Muslims, so it is certain. But the marriage may be checked in some ways. If 
    the woman is one whom it is permissible for him to marry in the first place, 
    the marriage is approved, but if she is one whom it is not permissible for 
    him to marry in the first place, such as a mahram (close relative) through 
    blood or otherwise, or because she is in ‘iddah, an apostate, an idol 
    worshipper, a Magian, or one whom he has thrice divorced, then it is not to 
    be approved. If they got married during the ‘iddah and they became Muslim 
    after the ‘iddah ended, then it is to be approved, because it is permissible 
    for him to marry her in principle. 

    End quote from al-Mughni, 7/115 

    Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said: 
    Marriages of the disbelievers come under the same rulings as marriages of 
    Muslims with regard to consequences such as zihaar, li‘aan, talaaq, 
    attribution of lineage, and so on. 

    Secondly: 

    If the marriage is valid according to Islamic sharee‘ah, then 
    it is valid. If it is invalid according to Islamic sharee‘ah, then it may be 
    approved subject to two conditions:

    (i)               
    that they regarded it as valid 
    according to their own laws

    (ii)             
    that they did not refer to us 
    for judgement. 

    If they do not regard it as valid, then they should be 
    separated. If they refer to us for judgement, we should examine the case. If 
    that is before the marriage contract has been done, then we should do it 
    according to our laws. If it is after the marriage contract has been done, 
    then we should find out whether the woman was permissible (for him to 
    marry), in which case we should approve the marriage, but if she was not 
    permissible then we should separate them. The evidence for these things is 
    the fact that disbelievers became Muslim at the time of the Prophet 
    (blessings and peace of Allah be upon him) and he left those who were 
    married on the basis of their marriages during the Jaahiliyyah and did not 
    discuss the matter. This indicates that (such marriages) are to be left as 
    they are.

    End quote from ash-Sharh al-Mumti‘, 12/239 

    Based on that, the children who are born to non-Muslims, if 
    they were born within a marriage that they believe is a valid marriage, are 
    indeed their children and attributing them to them is valid. 

    But if they did not believe that this was a proper marriage; 
    rather it was a kind of friendship and love without marriage, then these are 
    illegitimate children, not legitimate. 

    But if the zaani attributes them to himself and regards them 
    as his children, then they are his children and are to be attributed to 
    him. 

    Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) was asked 
    about the problem of zina (fornication or adultery) which few of our 
    brothers and sisters were free of before they became Muslim; and many 
    children resulted from these relationships. With regard to the children who 
    are born as a result of zina (fornication), is it obligatory for their 
    fathers who have become Muslim and who were not married to the mothers to 
    spend on their maintenance? 

    He replied: With regard to these people who had intercourse 
    when they were still disbelievers, if they believed that this intercourse 
    occurred in the context of a marriage contract that they thought was valid, 
    even though it is invalid according to sharee‘ah, the marriage contract is 
    valid and the children belong to that man. For example, when he was still a 
    disbeliever, a man proposed to a woman that he should be her husband and she 
    agreed, and they thought that this was a marriage contract, then the man and 
    woman became Muslim. We say: Your marriage is still valid and there is no 
    need to repeat the marriage contract, and any children that you had are your 
    children, unless the wife is not permissible for the husband in Islam. An 
    example of that is if he was a Magian and married his sister, as the Magians 
    regard it as permissible to marry mahrams (close relatives). If he married 
    his sister when he was a disbeliever, then he became Muslim and she became 
    Muslim, they must be separated because the woman is not permissible for the 
    man. With regard to the group of people you mention we say: If they believe 
    that what they did of having relationships with these women was a marriage, 
    then it is not zina and the children belong to them. But if you believed 
    that it was zina and you attributed these children to yourself when you were 
    still a disbeliever -- i.e., the zaani said: These are my children -- then 
    they are his children, so long as there is no one to dispute with him on 
    this issue. But if they did not attribute these children to themselves, then 
    they are not their children. 

    With regard to maintenance, if we determine that they are 
    their children, then they are obliged to spend on their maintenance; if we 
    have not determined that, then they are not obliged to spend on their 
    maintenance. 

    End quote from Fataawa Noor ‘ala ad-Darb.