There have probably been several rewrites, but similarities tie "The British Sergeant" to the 19th c. Britania's Disgrace. This is clearly shown by the 'grasshopper' verses.
I would like to see the versions in Fuson, even if they are fragments. They may help to place the time of origin.

Nothing at American Memory so far found puts the origin in the 18th c. "A Song Composed by the British Soldiers...." has nothing that allies it to "Sergeant," except that it purportedly is by soldiers.