I'm guessing you are on Scandinavian?
You don't have to re-check in in CPH, though you may need to go through passport control if you are transiting from an international to a intra-European flight. Not a big deal, they just ask where you are going and may ask if they see the visa in the passport. Your luggage will transit through to LHR without you having to touch it again once you drop it in Chicago (assuming you are on all flights with Scandinavian or a major carrier and arent changing to something like Ryanair).
If you have a long layover, it can be worth it to go into Copenhagen proper as the airport is practically in the city center. Eating at the airport is a massive rip off, though, so bring snacks if you want to make it to London without paying for food in Denmark (Scandinavian pricing is no joke).
On our move over we took SAS from Chicago to Stockholm and they 1) forgot to charge us for our kitties and 2) didn't charge extra for our extra bags, but we had found cheap premium economy tickets and had the extra bag included. Its a pretty stripped down service - one meal option in Economy and they board in an awful way whereby if you have no status you dont get a seat assignment until you are at the gate, and then you get stuffed into the middle of the airplane. I typically refuse to fly them transcon, but for the sake of our kitties I didn't want them to have to fly more than one leg. Just.... don't expect much service since it says Scandinavian on the side of the plane, although the CPH airport is incredible, clean, streamlined, and has a lot of conveniences most other airports don't have.
Finally - there is an outlet of Tortas Fronteras near the SAS departure gate at ORD - can HIGHLY recommend getting a torta there before boarding. That alone almost caused me to disembark and stay on American soil!