In my view it's resume'.

My reasoning is it is pronounced reh-zu-may (English speaking countries)
And the e' part is not because we are giving reference or respect to french history, but because the ending vowel changes its sound when it has an accent placed above it. ie Instead of resumee its resumay.
Which is resume'

That's how I was taught anyway.
But, I do notice my iPhone places both accents on, so that's a bit irritating.
I wonder what English teachers (or English professors) teach their students in school.