In Project Settings >> Input Devices >> Pointing, I have both Emulate Touch From Mouse and Emulate Mouse From Touch enabled.
That seems to produce the desired results. My game responds to mouse input on my desktop computer and to touch input on my Android phone.
One change I had to make for the phone was making buttons larger so that they could easily touched with a finger. In some cases I used a trick: I made the button background transparent, so that the icon or text could be small, while the larger button area responds to touch input.