Let me try to explain when the InputBindings will be fired. In a regular case in which you have window and some controls in, by default the focus is inside the Window, that is why if you declare an InputBindings for the Window they will be triggered. However in the case where you have UserControl inside the Window, the ImputBindings for the UserControl will not be fired, unless you focus it.
If you entire application is hosted in that UserControl you could focus it once the is loaded and the bindings should work as expected. Another approach would be to declare the commands in both the Window and the UserControl to the same command. Using this approach will fire the command as you expected,
I hope this information helps. Please let us know if you need further assistance.