The former Trope Namer is "God Save the King/God Save the Queen," sung in America as "My Country, 'Tis of Thee" and in Imperial Germany as "Heil dir im Siegerskranz."
Even funnier, one of the first Russian national anthems was this British anthem with new lyrics in Russian. But soon the British Empire became the rival to the Russian Empire, and Russians wrote their own anthem with their own tune ("Spasi bozhe tsarya khrani"), that survived until the revolution. It is in most American hymnals as "God, the Omnipotent, King Who Ordainest."
And after the revolution, things got even funnier. The Soviet anthem "Soyuz Nerushimy" (The Unbreakable Union) was originally, before WWII, an anthem of the Communist Party, with different lyrics. After Stalin died, the lyrics were at first totally removed under Khrushchev, and under Brezhnev the anthem received new lyrics, particularly changing the second stanza that hailed Stalin, and totally removed the third stanza that emphasized the military. And the current Russian anthem is basically Soyuz Nerushimy with lyrics changed once more again, now removing all references to Lenin and communism and replacing them with your garden variety patriotic lyrics.
Possibly Older Than They Think: the initial version of the theme was allegedly written by Jean-Baptiste Lully for the King of France Louis XIV, then plagiarised by Händel, who sold it to the British crown.
The national anthem of Liechtenstein is sung to the same tune. Hilarity Ensues when England play Liechtenstein in football.
And somehow somewhere in the mists of time since Lully and Händel, the tune entered Swedish folk music, where the key was changed to minor (Swedish folk music is nearly always in minor key). So in Sweden, it's a sad little song on how it's actually better to be abandoned than to be locked in a marriage with someone who doesn't love you, even if both of them suck.
The temporary Soviet anthem and worldwide Communist song "The Internationale" was originally sung to "The Marseillaise".
Communists in several Latin-American countries still sing it that way.
The modern version doesn't sound much like a drinking song because John Philips Sousa altered the tempo to make it sound more dignified. Modern people familiar with the Sousa arrangement might be quite surprised by earlier, jauntier versions.
The first Korean national anthem was originally a traditional Korean poem sung to the tune of Auld Lang Syne.
O Canada, the Canadian national anthem, was originally a patriotic French-Canadian poem set to music for a Saint-Jean-Baptiste Day ceremony. The English lyrics were written much later and do not constitute a literal translation. (Ironically, this is now the one song you are probably least likely to hear in Québec on Saint-Jean-Baptiste Day, due to that holiday's strong association with the sovereignist movement.)
Germany's national anthem uses a tune that was originally written by Joseph Haydn for Francis II of the Holy Roman Empire. The tune is also used for several hymns. Most Americans probably associate the song exclusively with World War II, but it's Older Than They Think.
The University of Texas' school song, "The Eyes of Texas", is set to "I've Been Working on the Railroad".
The Alma Mater Song of Yale is a taken from German patriotic hymn "The Watch on the Rhine", best known in America for its appearance in the "Marseillaise" scene in Casablanca.
"The Red Flag" was not originally intended to be "O Christmas Tree". For example: "Our Christmas tree is deepest red / it's shrouded of our martyred dead / and ere its limbs grew stiff and cold / our heart's blood dyed to every fold / so raise the scarlet standard high / beneath it's shade we'll live and die / though cowards flinch and traitors sneer / we'll keep the red flag flying here."
"O Christmas Tree" was originally in German, as "O Tannenbaum", which means pretty much the same thing — okay, literally, it's "O fir-tree," but what, you were expecting maybe a Christmas maple?
"Turkey in the Straw" (verse), and "Do Your Ears Hang Low". Also the Doughnut Shop song. And the Animaniacs state capitals song. One of the earliest incarnations of this tune was the minstrel song "Zip Coon."
"Have You Ever Seen a Lassie" and "The More We Get Together".
"The Itsy Bitsy Spider", "Sweetly Sings the Donkey", and "Ten Green Bottles".
"Johnny Comes Marching Home", "The Ants Came Marching" and "Ghost Riders in the Sky" all were based on the Irish ballad Johnny I Hardly Knew Ye
The satirical Taisho-era Japanese song "Tokyo Bushi" (often called "Pai-no-pai-no-pai" for its nonsensical chorus) took its melody from an American Civil War marching song, "Marching Through Georgia".
"Marching Through Georgia" itself had new lyrics set to the tune of the chorus called "Hooray, They're Hanging Father."
"This Little Light of Mine", and "Worried Man Blues".
Patsy Gallant gave us "From L.A. to New York", to the tune of Québecois folk song "Mon pays". The English lyrics (written by Gene Williams) are nothing at all like the French lyrics, and the original author (Gilles Vigneault) has disowned the English version. (to be fair, Mlle. Gallant also recorded the French version in the same disco style)
The martial, patriotic marching song "The British Grenadiers" is to the tune of a seventeenth-century comic song called "The New Bath".
"I Saw Three Ships" and "Here We Go Round the Mulberry Bush".
"For He's A Jolly Good Fellow" AKA "The Bear Went Over The Mountain" AKA "We Won't Get Home Until Morning" is known in French as "Marlborough s'en va en guerre."
Baracuda did this with using the tune from Amaranth by Nightwish, and the lyrics to "Where Is the Love?" by Alice Cooper. This can be seen here.
"Frère Jacques", "Where is Thumbkin",, the Chinese children's song "Liang Ji Lao Hu" ("Two Tigers"), and Binky the Clown's birthday song from Garfield and Friends.