You shouldn't have to worry; the amount of snow forecast for Thursday is nothing for O'Hare. I don't mean nothing as in zero snow, I mean nothing as in nothing significant from the standpoint of ORD (and Chicago and Chicagoans') operations.

(As to your backup plan, keep in mind, any weather that would seriously screw up or shut down ORD is going to seriously screw up or shut down MDW too, and probably before ORD. I would guess the same is true for Milwaukee.)

ORD is my home airport, and I've almost never in my life seen ORD operations seriously disrupted by a local weather event. A 10- or 20-year blizzard, maybe. But the snowstorm that is hitting the East Coast passed us by this time. And I imagine whatever delays that causes will clear out before Thursday.

As already mentioned, winter weather typically only affects ORD in terms of the domino effect of flight delays and cancellations at other airports (e.g., Boston, NY).