According to this statement here (taken from kotaku.com) the English speaking users seem to be the only one's being a bit screwed right now:
We are continuing to look into the issues people are experiencing with The Pitt for Xbox 360. As of now it looks like the file for the English version on Xbox Live was somehow corrupted. We are currently working with Microsoft to remove the existing file off of Live to prevent any further downloads. Our plan is to replace it with a new file as soon as possible. We hope to have another update for you shortly.
My (German) version works fine so far and I've already completed two quests, what about people using other versions like the French one, do your versions run fine or not?