Strictly speaking he would be a trumpeter - not a bugler. The infantry used bugles, the cavalry used trumpets. I assume this was following the practice in the British army.
Nearly.
The cavalry trumpeter [sic] used the trumpet in barracks and camp, but the bugle in the field .............. louder, lighter and simpler. Photos abound of the mounted trumpeter with trumpet and bugle.
So the order would be "Trumpeter!" but he would tootle on the bugle.