Thank you for your response.
I do apologise for my delay in responding to you. Thank you for your patience.
The reality is that your ex cannot take the children to America to live without your permission.
If your ex was to do so then a court would make an order for the children to be returned. Your permission is required and if you do not provide your permission then your ex would need the permission of the court.
If your ex was to take the children out of the jurisdiction without your permission then she could be charged with child abduction and if found guilty face a prison sentence.
If you get the impression that your ex is taking steps to remove the children without your permission then you should make an urgent application to your local family court for a Prohibited Steps Order.
If, you do believe her intentions are to travel at the end of the year and you object to the same then you can still apply to the court for a prohibited steps order but you would need to attempt mediation first as this would be considered by the court as less urgent then immediate plans to move.
If mediation is unsuccessful at reaching agreement then then mediator will sign the form that you need to apply to court. This is the C100 Form. The court fee is £215.
The mother, if she is serious at relocating, will likely ask the court for an order that she can permanently remove the children from the jurisdction.
This is not an easy application for your ex to make to court and the court would have to be convinced that the move was in your childrens best interests.
The court would ask a Children and Families Officer (Cafcass) to prepare a report on the matter and make recommendations. The court will want to know how the mother intends to promote contact and this will be by no means easy for her given the travel time to the USA and back. Cafcass will ask your childrens wishes and feelings but this will not be decisive given their ages.
In short - if you dont agree - do not provide you permission. The mother will have to ask the courts permission and this, given the recent case law, is not going to be easy for her.
Do let me know if I can assist you further.
Positive feedback is gratefully received.