The importance of British participation in strikes on the Houthis shouldn't be lost on anyone. This is not just your standard Anglo-American cooperative military effort.
The U.S. currently produces more oil than it consumes. This is even before you account for Canadian, Mexican and Caribbean production. The Houthis (and indirectly Iran) are flailing about at the wrong enemy. Europe is reliant on Mideast oil coming through the Red Sea and onward through the Suez Canal.
While these attacks aren't sponsored by NATO, I bet the NATO countries support these attacks. I don't know the time, coordination and politics needed to bring in say the French or anyone else. Let's just say the U.S. as the world power with the responsibility for keeping global trade routes open and Britain can usually get on the same page rather quickly.