All UK businesses will have to ensure they meet the requirements of the UK legislation.
The laws apply across the EU, although are implemented differently in each country. All businesses in the EU will therefore need to comply with the regulations, and will be bound by those in their own country.
Cookie Law Example of What businesses will have to show consent?
In theory, any business anywhere that has a website serving customers within the EU, is required to comply with the legislation with respect to those EU visitors. So a US website with UK visitors, ought to be asking for consent from those UK visitors according to the UK legislation.