I don't own Apple products, so I can't test the live casting for you. I think the issue is the mobile device has to use the NFL app (not a web page in a browser), and the NFL app restricts casting, but I am not sure. I do believe they offer a free trial though, so you can sign up right before the game and see if you can make it work, and cancel if it doesn't work for you.
For the re-watch games, which from what I understand are pretty much available as soon as the live game ends, I don't need to cast at all. I just use the NFL app on my TV.