After finishing episode 1, it brought me to the main menu. I selected my game file, hit "Play" on episode 2, and the game starts me at the very beginning of episode 1.
I've tried returning to the dashboard, and trying again, which doesn't fix it.
I have the full episode set on disc, this isn't a download issue! In fact, I don't even have access to an internet connection on my Xbox in the first place, so I really hope I don't have to connect to fix this!