It is very much the responsibility of the parents to decide the
religious upbringing of their own children. It is normal when
having a child baptised to have godparents who agree to assist the
parents in their duty, but the godparents can not do anything
against the wishes of the parents.
If for any reason, the parents have subsequently left their former faith, then they are unlikely to want their children brought up in the faith. In this case, the godparents have no further responsibilities of a religious nature, although they will probably remain family friends.
Choosing Godparents today doesn't mean much of anything with the exception of the parents regarding the couple highly enough to even ask them. Being a true Godparent years ago meant that the Godparents were there to help in any way they could and if both parents were to pass on then the Godparents were to take the child(ren) in as their own. Now it has simply become a ritual. If the child was baptized as a Catholic,then its the responsibility of the parents and the godparents to ensure that the child is raised a Catholic. During the baptism the Parents and Godparent(s) would make a VOW on behalf of the child.So if the Parents don't fulfill their vows or changed their faith or died then it would be the God parent(s) duties to ensure that their Godchild's baptismal vows is fulfilled...(by teaching them of their christian duties as a Catholic and being an example of faith and prayer etc) It doesn't mean that by being a Godparent you have a legal responsibility for the child,but you do have a spiritual Responsibility....thats why its called God-Parent...Then when the child's older they'll renew their baptismal vows through Confirmation..