In this case, 18. It will depend on if the father agrees before this age. The reason is, that a court is unlikely to allow a child to decide an action which will effectly take that child out of the jurisdiction of the US. If the children want to, and the father disagrees then he will probably win any change of custody case.
There are other issues here, once the children are gone, there is nothing the father can do in a dispute. Sweden do not recognise any foreign custody orders so the father will have no chance in the future. Once the children have resided in Sweden for 6 or more months, the mother can file for full custody and take the father completely out of the picture..