I was wondering about this because I don't want to get in trouble when it comes to taxes. I live in the United States, in New York.
If you are a U.S. citizen or permanent resident, you must pay U.S. income taxes on your gambling winnings anywhere in the world (see generally multiple resources from the IRS). But, you may take a credit against the U.S. income tax liability on those winnings for any foreign (e.g. U.K.) taxes you pay on those gambling winnings.
It does not matter if you bring the money back or not. Repatriation of earnings is a concept that applies to foreign subsidiaries of U.S. owners, not individuals.
All of this is assuming that no U.S.-U.K. tax treaty applies. There are two relevant treaties, one from 1975 and one from 2001, each of which is explained in greater detail than the bare treaty language approved by the U.S. Senate in technical explanations and protocols reached between the respective tax collection agencies. I am not aware of either that applies to gambling winnings but I am not an expert on these treaties and you'd have to read them to be sure.
There is an exemption for foreign earned income of individuals, but gambling winnings are not earned income unless gambling is your occupation rather than merely entertainment or an occasional activity.
But, you would not owe New York State or New York local income taxes on those winnings.
Of course, while U.S. gambling establishments report gambling winnings to the IRS, those in the U.K. might not, which doesn't mean that you don't owe the taxes, but reduces your odds of being caught if you do not.