While Edison's style remains unacknowledged as a vertical style in books on the history of penmanship, it clearly is upright, unlike the style of the time. However, it is fair to say that he neither authored a book on penmanship nor taught it in any institutions, and so he cannot be seen as among those who influenced the nation's youth nor those outside of telegraph offices. 

In chronological order, one can point to the next iteration of a vertical style that was promulgated in Europe and later came to be popularized by John Jackson in England in the 1880's. While it is hard to find the exact date that Jackson's vertical style came to be known by those in the United States, there is evidence that it was not later than 1887. Here, one can see an advertisement of it in The Educational Times

Edison's style and news of his efforts did, however, reach Melvil Dewey, who was perhaps inspired by Edison. Melvil later introduces "library hand," itself a vertical style, as the official style to be used by all librarians. Like Edison, he was interested in speed, efficiency, and legibility. One can think of the vertical style as the handwritten equivalent to the typewriter, which was rapidly evolving as a fast and efficient office machine that created its own vertical print. 

However, likely neither Edison nor Dewey would have probably approved of the evolution of their vertical script ideals. In many writers, vertical script evolves into a backhand style. Here is a library card written for a gift received in 1909. This particular script has the worst qualities of being somewhat ornate, which Edison had avoided; it is also not in Dewey's prescribed style. And of course, it is harder to read since it is both slanted, and slanted backwards. Oh, progress!