ABBA could write a book (or a song) about losing one’s native accent while singing in a different language.
Sometimes, one cannot get rid of the accent, or doesn’t even try. In the song 99 Luftballons, I could clearly hear Nena’s German accent. I thought it added to the “Worldly Gypsy” credibility of that song. That quality helped it to sell.
You’re right, if you pay attention you can still hear their accents, but they often try to disguise it so as to reach a wider audience.
Once Tom did Southern Accents, he let his Southern twang loose more often. Made it easier for me.