BIG warning to you: the father needs to approve of your taking your child abroad and custody re-structured accordingly. Being in active military service does not constitute 'unavailable'. He's available, just harder to pin down.
Even if he is unaware he can stop it now, he can find out from any JAG attorney, especially if he starts asking questions about the amount of child support he pays or visitation. And then, you're in trouble, potentially serious. The UK and US are part of the Hague Agreement regarding child custody. Taking your child abroad w/o the appropriate consent can be seen as abduction, and believe me, it's treated as a crime in the courts both here and in the US.
Lots of parents on this board have worked with their former spouses and partners to bring their children across legally and made appropriate custody arrangements. It can be done! But do yourself a big favour and don't try to 'loophole' your way out, b/c there really aren't any loopholes unless the other parent is completely untraceable.
A post in the parenting section could bring LOTS of helpful suggestions.
Good lucky!