Presumably you refer to AppleCare+, this being the optional extension to your device warranty that provides coverage for accidental damage. Irrespective of title ownership, be aware that AppleCare+ plans have territorial limits.
Typically, a claim under an active AppleCare+ plan can only be made in the country/region that the plan was purchased. An AppleCare+ plan purchased within the UK will only be honoured within the UK. Similarly, an AppleCare+ plan purchased within the EU will be honoured within any EU Member State (noting that the UK is no longer within the EU). This limitation applies equally to AppleCare+ plans purchased with a new device within the US.
As such, you will be unable to access repair/replacement services relating to your UK AppleCare+ plan while travelling in the United States.