Praise be to Allah.
Your wife’s brother can assume the (wali) position. If there is not a brother then an uncle would do. He second eldest son would also do, if he is an adult. In the absence of all of these, an Islamic judge or the head of the Islamic center can renew the contract. You don’t have to tell anyone about the reason. You can say that you just want to renew it because you don’t feel good about the first one. Publicity is not needed at all. There was an addition to my answer to you, but apparently it was lost for one reason or another. Here is the full answer once again:
The marriage contract is void because it was made without fulfilling its necessary conditions. One of these conditions is the readiness of the womb. This means that the woman whom is to get married must have her womb unoccupied. For example, a man may not marry a woman who is pregnant. He may not marry a woman who was divorced until she is out of her idd’ah (a period where a widow or a divorced woman may not marry). Also a man may not marry a woman he has been having intercourse with until they both repent and she gets her monthly period. This is a sign that her womb is clean. The Prophet forbade Muslims to have intercourse with female slaves they bought recently until they are certain that their wombs are clear from any pregnancy. Waiting for the monthly period does this. In your case, the marriage contract must be renewed. This is not a difficult thing to do. The first boy is not Islamicly your son, as he was not created in his mother’s womb through a valid marriage contract. Some scholars say he is not to be called after you as he is born out of adultery. He is to be called after his mother’s family. He is not to inherit you and you are not to inherit him. As for the remaining children, they are yours and they carry your name. Other scholars, such as Ibn Taymiyah and Ibn AlQayyim say that if the woman is not married at the time of committing adultery, then the adulteress father may give his name to this boy and that he may be treated as his son. In your case, this last opinion may be the most suitable for you and your family. Yet you must renew your marriage contract at any Islamic center, and Allah knows best.