The mother needs an attorney, ideally one licensed in both SC & NY, ASAP!
Generally speaking, if a child is born out-of-wedlock, the biological father has parental rights equal to the biological mother once the father has been determined to be the father, absent any steps to take his parental rights away through the courts. That can happen through signing the birth certificate and/or a court order establishing paternity. You need a court to either decide, or approve of an agreement deciding, (1) legal custody and (2) physical custody. If the mother never got a court order saying that she was the sole legal custodian of the children, and the father had no legal rights to the children, then the default would be they both have legal custody. If she DOES have a court order saying that she has sole legal custody, and the father does not, (and it also seems safe to assume that the father wasn't following the physical custody plan when he took the kids), the father kidnapped the children. The mom needs an attorney at this point to figure out how this will proceed and what rights she and the father have or don't have. An attorney will also help her figure out whether the NY or SC courts have jurisdiction over the children at this point depending on their residency status.
Often times a judge will grant sole legal custody to the mother of a child born out-of-wedlock, but certainly not always. I don't know if that was the case here, but the sooner a lawyer can help this mom, the better. The clock is ticking with respect to the children and how a court will view what is in their best interests. Court orders with respect to children can be modified (changed) and who knows what the father has done in the meantime with respect to trying to get legal & physical custody.
This post is for informational purposes only. You should not construe it as a legal opinion on any specific facts or circumstances, and you should not act upon this information without seeking professional counsel. Publication of these materials and your receipt of them does not create an attorney-client relationship.