"Heil Dir in Siegerkranz" ("God Save the King/Queen"/"My Country 'Tis of Thee") is one of the Great Sources of Parody. (On my list, it's co-#17 in most-parodied. The top five: "Yankee Doodle," "Packington's Pound," "John Brown's Body," "Fortune My Foe (Aim Not Too High)," and "Auld Lang Syne.")

The Ballad Index lists the following to this tune, and this isn't even close to comprehensive:

O Deus Optime (cf. Chappell/Wooldridge-OldEnglishPopularMusic II, p. 195)

South Carolina, A Patriotic Ode (File: CAFS1298)

My Country (Greenway-AmericanFolksongsOfProtest, pp. 88-89)

God Save the King (The King He Had a Date) (Pankake/Pankake-PrairieHomeCompanionFolkSongBook, p. 115; see next item)

Poor Georgie ("Queen Mary, so they say, Had a dictating way"; this appears to be a version of "The King He Had a Date" customized to George V and his wife Mary of Teck) (Harbin-Parodology, #47, p. 18)

My Country's Tired of Me (Pankake/Pankake-PrairieHomeCompanionFolkSongBook, p. 159)

Can Opener, 'Tis of Thee (Pankake/Pankake-PrairieHomeCompanionFolkSongBook, p. 159)

Our Land Is Free (celebrating the end of transportation to Van Diemen's Land) (Robert Hughes, _The Fatal Shore_, p. 572)

God Save the Rights of Man (1798 Irish revolutionary song) (Lawrence-MusicForPatriotsPoliticiansAndPresidents, p. 128; mentioned in Thomas Pakenham, _The Year of Liberty_, p. 193)

God Save America (Lawrence-MusicForPatriotsPoliticiansAndPresidents, p. 78)

His Excellency George Washington ("From the Americ shore, The vast Atlantic o'er, Shout -- 'Washington!'") (Lawrence-MusicForPatriotsPoliticiansAndPresidents, p. 92)

(Washington songs) ("Americans rejoice, While songs employ each voice" and "Hail Godlike Washington! Fair Freedom's chosen son") (Lawrence-MusicForPatriotsPoliticiansAndPresidents, p. 96); Ode to be sung on the Arrival of the President of the United States ("Hail thou auspicious day! Far let America Thy praise resound") (Lawrence-MusicForPatriotsPoliticiansAndPresidents, p. 117)

Ode ("Now let rich Musick sound And all the Region round With Rapture fill") (Lawrence-MusicForPatriotsPoliticiansAndPresidents, p. 97)

Ode for the Fourth of July ("Come all ye sons of song, Pour the full sound along") (Lawrence-MusicForPatriotsPoliticiansAndPresidents, p. 128; Rabson-SongbookOfTheAmericanRevolution, pp. 74-75)

Closing Ode ("Patrons, with dignity, The basic industry") (by A. P. Knapp) (Albert P. Knapp, _Grange Songster_, 1915, p. 3)

So Say We ("So say we all of us, So say we all of us, So say we all") (Rodeheaver-SociabilitySongs, p. 125)