Is a UK trademark valid in the US?

A UK trademark does not extend to the United States, as trademark protection is generally territorial, meaning it applies only within the country or jurisdiction where it’s registered. To obtain rights in both the UK and the U.S., therefore, you will have to apply for its registration in each of the countries separately with their respective trademark offices: for the UK, the United Kingdom Intellectual Property Office, and for the U.S., the United States Patent and Trademark Office.

Contrast that with the Madrid Protocol, which is intended to make applying for trademarks across borders less complicated. Through this system, under the auspices of the World Intellectual Property Organization, you'll be allowed to seek protection in multiple nations with one application. In actuality, though, each country would still examine an application and grant the status of the application, including approval, based on its own laws. This would mean that, even though one submits a single international application, the registration and protection would, therefore, remain under different requirements and regulations of the chosen country.