Just to clarify to others - the Mother ALWAYS has the right to relocate. Human rights law that she can live wherever she wants. So the argument is - whether or not the child should be relocated. This comes down to - who does the child live with. So it was probably always going to be a case of - which parent is the resident parent, and which one visit - if the Mother decides to go abroad.Please try and appeal this decision. You’ll get a different Judge.
However in one Uk relocation case, the Judge made an order for the children to live with the Father and said it was then up to the Mother if she still wanted to move or not and if so she'd have to travel to see the children. Obviously travel in the Uk is a lot easier and more frequent than the other side of the world.
But that is what the Judge should have been looking at. Not deciding whether the Mother could relocate, but deciding who the child would live with "if" the Mother decided to relocate. It should have been residency with you. Then your ex could decide whether to get another job in the Uk or decide to go back to the US.
As you say - there are probably many grounds for appeal - it was a final hearing, not a fact find, for a start off. Take a few days to get over this awful experience and then think.
Although of course your ex might jump ship before you get chance to appeal so it would probably need doing quickly.