There is no way Britain can become a U.S vassal state, what has happened is that Theresa May is seeking to rekindle the once tight ties that existed between Landon and Washington. The idea of Britain becoming subservient to United States were given by French presidential hopeful Emmanuel Macron, who is among people who thought Brexit marked the end of Britain, that it will never find another partner.
Britain is not becoming a U.S. vassal state. While the country has provided much allied help in the past, it is an independent nation on its own and provides support to many countries in time of need. It has a strong military and government that attends to many affairs around the globe.
The United States is the dominant partner in the Anglo-American alliance and has been since 1956 when the British tried to seize control of the Suez Canal from the Egyptians and the American president, Eisenhower, threatened to destroy the pound on international money markets if they didn’t back off. On the other hand, the two nations have been sharing intelligence data with each other since 1940 and that “special relationship” makes them an equal partners than anything else. The ability of the United States to dictate British policy in any area in any substantial is close to nil.
Theresa May received flak from her own people for getting chummy with Trump, so I don't see how they are going to allow Britain to be subservient to the US. The British did vote for Brexit so I don't think they'll want to pursue a relationship that puts them in a lesser role anymore.