From the recording Jefferson and Liberty
The melody of "British Grenadiers" appeared in the revised version of the burletta pantomime "Harlequin Everywhere," "which reopened in January, 1780 at Covent Garden, after the Americans had been bloodily thrown back from Savannah, Ga., during the War of Independence" (Winstock, 1970). However, another version exists from 1745, and parodies of that time indicate it was considerably older than that (Winstock, 1970). The subjects of the title, grenadiers serving in the English army, were originally soldiers who threw grenades "and thus tended to be long in arm, big, tall men," according to historian Byron Farwell (1981). Grenades went out of fashion for some time in European warfare, but grenadier companies consisting of the tallest men were usually attached to battalions and were thought of as specialized, somewhat elite troops, so that "...by the First World War the term 'grenadier' had so changed its meaning that when the grenade throwers returned to the battlefield there were objections to calling them grenadiers and they became known as 'bombers.'"
"La Belle Catherine" is a French fife tune, and is reputed to have been played during the Battle on the Plains of Abraham, one of the last decisive battles of the French & Indian War. It had become a very popular fife and fiddle tune by about 1780 and over time, "La Belle Catherine" turned into the dance tune now known as "Come Dance and Sing."