All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
Any marriage without the consent of the guardian of the bride and the presence of two witnesses is invalid according to the correct opinion of the Muslim scholars. Any marriage that lacks a guardian and witnesses is null according to the view of the Four Schools of jurisprudence.
The Hanafi School is also of the same opinion. Al-Mabsoot authored by the prominent scholar as-Sarkhassi reads: “It is a condition for the validity of the marriage that the witnesses hear the words of the two contracting parties, i.e. the bride and bridegroom.”
This text proves two matters:
1) The presence of two witnesses is a compulsory condition.
2) Hearing the words of the bride and bridegroom is also a compulsory condition.
One witness is not sufficient according to the scholars who stipulate the presence of two witnesses.
Therefore, the marriage was invalid and the husband and wife should be separated. If there are any children born from this marriage, they will be traced back to the father. Shaykh Ibn Taymiyyah said: 'All Muslims are agreed upon the fact that every child that comes to existence from a marriage that was considered valid by the husband, such a child is traced back to the husband and they inherit from each other even though the marriage is invalid in itself.'
Now, if they want to continue their marriage they have to renew the marriage contract fulfilling all the conditions of a valid marriage.
Know that the consent of the guardian is also a compulsory condition according to the correct opinion of Muslim scholars. The Prophet said: “There is no (valid) marriage except with a guardian.” [Abu Daawood, Ahmad and others].
If the guardian of the girl is absent, then she should wait if he can attend the marriage; otherwise, he may deputize someone to marry her off or the bridegroom may deputize someone in the country of the guardian to accept the marriage on his behalf.
If the girl does not have a guardian, then the Muslim ruler is a guardian for whoever does not have a guardian.
In non-Muslim countries, the person who is leader of the Muslim community or the one who settles their disputes is considered a guardian.
Allah Knows best.