If possible, I'd try to warp it using a little water, to get the rounded shape:
1. Set up some books or blocks in roughly a circle or square shape, so that your cardboard circle is supported off the workspace around the edge, like a table. The books should be all about the same height
2. Dampen the cardboard slightly with a wet paper towel. You don't want it waterlogged, but you want it to be softened a little.
3. Place the cardboard circle on the ring of books, and place a weight of some form (maybe a smaller book) in the middle of the circle. This should cause the cardboard to sink in the middle, creating a Captain America-like disc. You only want a slight curve, too much weight in the middle or two much height on the supports might cause the cardboard to fold or buckle.
4. Let it dry completely so the cardboard retains it formed shape.
5. Repeat with the other layers of cardboard, then glue together after they've all been formed.
(One other method is to use 1" foam mats, and instead of wetting the shield, you heat it slightly so the foam becomes floppy/flexible, then weight and form it like before. The obvious advantage is that if you use 1" foam, it saves you the trouble of forming 4 cardboard discs if the cardboard is the standard 1/4" thick)
As you glue the discs together, I'd turn each disc 45 degrees from the previous one, so that the corrugation within each piece is in a different direction (so one disc each at 0, 45, 90, and 135 degrees). The corrugation makes the cardboard only bend one way (you're two creases are bent in the same direction as the corrugation), so by making the corrugation of all the layers in different directions, it'll never be able to bend along one disc when the other 3 are resisting it. This will make it incredibly strong. For the top layer, make sure you measure out even widths for the creases, and use a straight edge to create a truly clean and straight crease.
Once you've got 4 layers for the wooden part of the shield, I'd make a 5th layer for the metal ring along the edge. Then, I'd cover the exposed cardboard on the edge with cardstock or posterboard, to cover the visible corrugation. Some masking tape or paper mache to cover the corners so the posterboard doesn't peel.
I would used small wooden discs or wooden buttons painted silver as the giant rivets along the edge. You can find them easily in the wood crafting aisles of Michaels or JoAnns.