If St. Louis is out due to the lack of direct flights, then have a look at flying in and out of Atlanta. There are direct flights both ways between Heathrow and Hartsfield. They are a bit more costly than what you said you could get by using O'Hare as your point of entry and departure and, of course, I don't know that you're planning on using Heathrow as you airport in the UK.

As Michael pointed out earlier, every day you don't spend simply driving from Point 'A' to Point 'B' saves on the cost of the driving portion of your trip. Using Atlanta rather than Chicago would save you roughly 500 mi (800 km) in driving and would still let you hit the Great Smokies/Pigeon Forge, Memphis and Nashville. But whatever you decide to do, I would do it soon. Airfare prices and lodging costs tend to rise in price as time progresses. That is especially true this year everybody seems to want to get back to travelling after the COVID shutdowns.

AZBuck