"God Save the King" has been around since the mid-eighteenth century, and it's likely that some British soldiers would have known it. However, it is more likely that other popular British patriotic songs would have been well known as well. Specifically, there is a patriotic song called "Britons Strike Home", which was written by Henry Purcell as an aria for his opera Bonduca about the Iceni queen Boudica and her revolt against the Romans. It was apparently more popular than most patriotic songs at the time, as it was sung on the deck of the HMS Tonnant before the Battle of Trafalgar, on the floor of the House of Commons upon the Prime Minister announcing war with France in 1797, and was even used by British loyalists during the American Revolutionary War.

While the melody has likely stayed the same, some lyrical differences have been recorded. The original lyrics read "God save great George the king" as you have noted. However, Ludwig van Beethoven was fond of transcribing folk songs, and published collections of British and Irish folk tunes. In his transcription of "God Save the King", the first line reads "God save our Lord the King" rather than the modern "God save our gracious king/queen".