This is really an international law question, not an immigration question. You could run afoul of the Hague Convention on International Child Abduction. You should consult with a family law attorney who is experienced in international child custody issues BEFORE you take your child about the country. I've added those practice areas to your question.
This is really a question for an lawyer that deals with immigration issues. I have added immigration to your category. Also I would call the Embassy for the UK and ask them if there is someone there that can answer your question. It would also be a good idea to get a formal custody order in place.
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This is not an immigration law question, really. Not US immigration law question, anyway.
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