I know satirical versions of songs are a dime a dozen ala Weird Al but are there a lot of songs recorded, either by the original artist or someone else, that changed the lyrics into a different song but not done for humor?
The best example I can think of is Elton John’s Candle in the Wind re-written as Candle in the Wind 1997 as a tribute to Princess Diana rather than Marilyn Monroe.
Another I know of is an alternate version of Suzanne Vega’s Tom’s Diner called Waiting at the Border written about the Gulf War.
Austrian singer Falco had a song in 1981, “Der Kommissar,” which was primarily in German, with a splash of English; it was covered in 1982 by the British band After the Fire, with lyrics which were primarily in English, with a splash of German (though the theme, about drug use, was unchanged).
However, for purposes of this thread’s topic, the song then was recorded again, this time by Canadian singer Laura Branigan in 1983, but with a different name (“Deep in the Dark”), and mostly different lyrics (though the melody and arrangement was similar to both versions of “Der Kommissar”).
Billy Bragg updates his lyrics in his live shows (they are full of 1980s political references). There was some minor controversy when he updated the lyrics to Sexuality to be more trans inclusive rather than just anti-homophobic:
Also my four year old watches the kids cartoon Beat Bugs which someone got hold of the rights to the Beatles back catalog (but didn’t have money to spare to pay writers who don’t suck ) The sanitize a lot of the songs, something she immediately caught onto when I added the originals to her car playlist (“They are singing these songs funnily!”)
I think the term you might be looking for is contrafactum.
Would something like “My Country Tis of Thee” and “God Save the King/Queen” count for the purposes of the OP? What about foreign tunes? “It’s Now Or Never” by Elvis/“There’s No Tomorrow” by Tony Martin/“O Sole Mio”?
There is also I’m sure a lot of examples of replacing a place name with the one they are singing in. e.g. Iggy Pop, as recounted in the book Trainspotting, sang “'Scotland takes drugs in psychic defence” in place of “America takes drugs in psychic defence” when he played in Edinburgh:
The original Japanese lyrics to Sukiyaki by Kyu Sakamoto, which hit #1 in one U.S. in 1963, were about a condemned prisoner. In 1981 A Taste of Honey released an English language version that reached #3, which had new lyrics about a failed love affair.
Many examples from blues. A prominent one is “How Long Blues” from 1928 by Leroy Carr, the melody of which was used two years later by Ma Rainey on “Daddy Goodbye Blues”.
Blues and country started out as folk music forms, in which nobody owns the music. So it was perfectly acceptable to reuse melodies and other elements of songs. This changed as folk music gradually morphed into pop music.
The one-hit wonder group Gun Hill Road recorded “Back When My Hair was Short,” about life in the 60s. It had some lyrics that mentioned drugs, so they rerecorded it with new lyrics without the drug references, and setting the nostalgia for the 1950s. The revised version became a minor hit.
Make 'Em Laugh from the movie Singing in the Rain was the Cole Porter song Be a Clown with new lyrics. It’s not known why Porter didn’t sue. He would have won hands down.
The U.S. Civil War song “When Johnny Comes Marching Home” has been being repurposed ever since. I know it primarily as “The Ants Go Marching”, but Wikipedia has a long list of variants and recordings.