The Royal Navy usually has a ship or two hanging around the Caribbean, both to provide military support to the dozen British Overseas Territories in the region that the UK is responsible for defending, and to provide support (including search and rescue) to the various Commonwealth Realms and Republics, and other countries that the UK still has strong ties to (there is often a destroyer or amphibious assault vessel in the area during hurricane season).

The article says this took place near Dominican Republic, as part of a join operation with the US Coast Guard. The Dominican Republic is between Puerto Rico (part of the US) and the Turks and Caicos Islands (a British Overseas Territory), so it may be that the US Coast Guard began a pursuit in their waters and asked the British ship to help out.

https://en.wikipedia.org/wiki/Maritime_Drug_Law_Enforcement_Act

The United States has enforcement authority over any ship trying to smuggle illegal drugs into the US. Any vessel that isn’t registered to any particular nation is subject to seizure or destruction. The Royal Navy recognizes this law as well. They work closely with the US Navy in the region because of British Commonwealth countries. The US Navy and allies are the ones who enforce the concept of international waters.

For example, The CCP doesn’t recognize the Taiwan strait as international waters. But we do.

See also Economic Exclusion Zone (EEZ) and Air Defense Identification Zone (ADIZ)