I'm surprised that no one mentioned "Lili Marlene," the music for which might be older than WWI, but the lyrics, I think, were written around 1915. Wikipedia mentions that and seems to indicate that it was a WWII song more than one for WWI.

It might not be a "trench song" sung by the troops, but there is a song from the era called "OUI, OUI, MARIE." It was sung in the 1947 movie, "When My Baby Smiles at Me," which I saw so many times that I memorized the lyrics, which are very much like this:

Oui oui, Marie
Will you do this for me?
Oui oui, Marie
Then I'll do that for you.
I love your eyes, they make me feel so spoony
You'll drive me loony
Stop teasin' me.
Why don't we parlez-vous
Like other sweethearts do?
I want a kiss or two
From my cherie.
Oui oui, Marie
If you'll do this for me
Then I'll do that for you
Oui oui, Marie.

Oh -- I just googled the song and, among others, this page came up:
   http://www.firstworldwar.com/audio/ouiouimarie.htm