Christian fundamentalism always was a specifically anglophone phenomenon: it did not "become" so. I don't think there's much of a direct link between the religion of the "zealots" who "left Europe" and Christian fundamentalism.

The original name for Christian fundamentalism was premillenial dispensationalism, and that is still the name of its theology. Premillenial dispensationalism—which must be seen as a new religion, an off-shoot of Christianity, like Mormonism—was invented by the Anglo-Irish evangelist John Nelson Darby at the end of the 19th century. (The emigration of Puritans from Europe to escape established churches occurred centuries before that, of course.) I believe that the main reason why premillenial dispensationalism took root in America wasn't the Puritan influence on American Christianity, but rather the lack of an established church, which would have suppressed this heresy.

A bomb, H bomb, Minuteman / The names get more attractive / The decisions are made by NATO / The press call it British opinion -- The Three Johns