For example, if I'm on Week 7 in one mode, do I continue from that point in the other mode or does it start from scratch?
I'm getting attacked and I don't want to risk not meeting the challenge by making it harder on myself and switching to Survivors.
Yes, it carries over. Your progress in the meta game is consistent no matter which game mode you play - you'll notice that your meta game progress (your week, day and challenge, if applicable) is visible from the first multiplayer menu, before you choose a game type.
