That can be cutting it a bit close. It works in your favor that the Hiawatha tends to be pretty punctual. In terms of your baggage, Amtrak will transfer it to your next train (as an airline would if your itinerary has a connection). Once you hand it over in MKE, you don't have to worry about it until you arrive in NOL.
Are you in sleeping accommodations on the City of New Orleans, or in coach? If you are in a sleeper, you get complimentary access to the metropolitan lounge in Chicago. When you get off of the Hiawatha, head straight there and the Amtrak agents in the lounge will take you out to your train. If not, you can flag a red cap down and request assistance. They will help with whatever carry-on bags you might have, and will lead you to your next train (be sure to have a few bucks for a tip if you go this route).
Your connection to the CONO is guaranteed, which means that if 340 is late, and you miss your train, Amtrak will rebook you on the next day's train, and put you up in a hotel/pay for meals. Obviously this isn't an ideal situation, but Amtrak will take care of you if the connection is missed.
It really comes down to your comfort level. Have you been to Chicago Union Station before? How well do you know it? It is much smaller than an airport (you can walk to any of the other tracks within 5 minutes). How well have you and family done in similar situations before?
If there's any significant doubt in your mind, I'd book the itinerary with the larger amount of connection time, but at the same time, assuming that the Hiawatha isn't late, 50 min certainly is doable.