Yep, a common anecdote from European science conferences is that by the second day, everybody do settle into the thick, averaged Spanish/German/French/Italian/Russian accent of their English which is pretty much equally understandable to everyone present except from the actual guys from Oxford, England.
Exactly! When I have to speak with actual English people, I do try my best to imitate a Americanised, TV show accent. When I speak to non-native speaker, I don't try and let my french go through. It's easier for everyone.