You can generally trace many open American style foxtrot figures to things that Fred Astaire and Ginger Rogers did in various choreographic sequences in their movies. And because of their movies there was probably more influence of American style on English style than vice versa. It wasn't until the 1950s that English style really started catching on internationally because the British Empire still had influence. It came to be called International style. Between then and now, there has probably been more influence of International style on American than vice versa, but it's been influence on an established style, not a derivation.
“Dancers believe that in a time of deceit telling the truth may be a revolutionary act.”
Most British "documentation" had been heavily political in character. It was not simply descriptive, but also prescriptive, largely and primarily in the interests of the English dance teachers involved. They established a power base formed by having organized the first extensive dance competition system which they exported around the British Empire and much of the rest of the world. They then created, a "Viennese" Waltz with virtually no steps in it, over the strong objections of the Viennese masters and a Slow "English" Waltz with an extensive syllabus.
"Somewhere Over The Rainbow" by Bruddah Iz
They created "Jive," based on an American dance, and wonder why Americans don't conform to their "descriptions." They created Tango, which is quite different from the Argentine version it was based on. America was unusual in that it had its own active dance "documenters" who did more or less the same thing in the U.S. that the English did in England. Except that the Arthur Murray and the Fred Astaire schools differed somewhat in their interpretations of the American Style. With that clash, the U.S. became much less influential in the dance world and the style didn't spread very much. The English simply enforced their standards more broadly, they mostly decided who won the "International" competitions, after all.
"We can't always choose the music that life plays for us,
but we can choose how we dance to it."