Phumphering thru my desk drawer, I found a list I made years ago of some US standards with music written by European (non-British) composers. There's no reason not to believe that there are similar gems "out there" for the finding now. I've listed the "hit" versions; they've all been widely "covered" as well.
You Don't Have To Say You Love Me (Dusty Springfield)
I Will Wait For You (many versions, maybe Andy Williams the biggest)
Let It Be Me (Everly Brothers)
A Man Without Love (Engelbert Humperdinck)
It's Impossible (Perry Como)
All Alone Am I (Brenda Lee)
The Way Of Love (Cher)
I Want To Be Wanted (Brenda again)
Yesterday, When I Was Young (Roy Clark)
Love Me With All Your Heart ("The Other" Ray Charles and his singers)
It Must Be Him (Vikki Carr)
The Way It Used To Be (Engelbert again)
Love Me Tonight (Tom Jones)
What Now My Love (Sonny and Cher)
The German orchestra leader Bert Kaempfert wrote the melodies that became Strangers In The Night (old Blue Eyes,) Spanish Eyes (Al Martino,) L-O-V-E ("L is for the way you look at me...") (Nat "King" Cole,) and Danke Schoen (Wayne Newton.)
I recently found a used Spanish-language Christmas compilation CD on RCA Victor with a song I think would be a terrific Xmas number with an English lyric. The arrangement's great too (gorgeous and surprising strings); wish I could give credit where it's due there. Here's a You Tube link:
https://www.youtube.com/watch?v=PAmOx61k2Vg