Legal Question in Family Law in Florida

You will have to have the case transferred to Florida.

It depends what court issued the Final Judgment. If the Alabama court took action, and the case was not transferred to Florida Alabama may hear it. If the child has been in Florida or Texas a significant length of time (usually 6 months or longer) then that state may hear the case.

You and the kids live in FL so the case will have to be registered and modified in FL.

First, there is almost no chance you'll have this expedited. Next, if the children primarily live with you, and you've lived in Florida for more than six months, your case will be litigated in Florida. Last, if you are permanently moving the children to the U.K., you need more than just passports. You need an order changing your timeshare ("custody") schedule. That will be extremely difficult if the other parent is following her timeshare ("visitation") schedule. Judges don't like to move children so far away from a parents that the parent will almost never see the child. If your ex is seeing the kids according to the schedule (or pretty close to the schedule), and she won't agree to the move, expect your case to be difficult, and expect the litigation to last months. Read the statute and follow it to the letter (or better yet, hire a lawyer). There are extreme penalties (like loss of custody) for moving the kids without getting a proper court order.