I believe that the English action was a descendent of Cristofori's action, while the Viennese action came from a separate invention of the piano by a German inventor; I cannot remember who offhand, but probably the piano maker whose piano Bach tried.
The Viennese action is not a repetition action. It is difficult to regulate and service. So the Erard repetition action has replaced it in modern pianos.
I really don't think it is fair to separate Cristofori's action from the Viennese school, BDB. Yes, with 21st century eyes we see the shortcomings of the Viennese design, and to modern American piano techs. it is counterintuitive that the hammer head should point toward, rather than away from, the player. It is simpler than a modern action by far, and is simpler than Cristofori's last action, but gave a quicker response and a somewhat louder sound than other early pianos of the time.
I feel better thinking about it as a step in the evolution of the action. It was also a great step forward at that time and offered new sensitivity in the late 18th and the first half of the 19th century. Mozart, Hummel, Beethoven, and others owned them and had great praise for them.
By the mid 19th century, the Viennese action was being out performed by what would become the modern action, and BDB nailed the designer. We can thank Erard for helping to point all action builders to where they are today. But the Viennese action still had demand and, until 1909, BÃ¶sendorfer offered it as an option in their pianos.
My 2 cents,