If you miss your connection to Chicago because the flight from Paris was delayed, EU261 has nothing to do with that. You will be re-booked on a different flight to the US, and if it is the following day your hotel should be covered.
We experienced this effect on our flight home last October, although it did not actually affect us personally. Our from Milan to London was delayed by a combination of circumstances: boarding was chaotic (too many people taking oversize bags aboard, not enough volunteered to gate-check, the overhead bins filled up and the aisles were blocked by people dealing with that. While we were waiting to get to our seat in the back of the plane, the pilot came on and said if everyone wasn’t seated within 10 minutes we would lose our landing slot at Heathrow, and would have to wait probably 2 hours for another. We all did get seated, but then there was another problem; a passenger who had checked bags could not be found. The pilot explained that we could not take off with that baggage aboard unless the passenger was on board. After another 10 minutes of sitting on the tarmac, the plane moved into take-off position and we thought all had been resolved. But instead of taking off, the plane taxied slowly all the way down the runway and turned back toward the terminal. We did not connect to a gate but just sat on the tarmac again for 20-30 minutes. We could not see what was going on, but I assume they removed the offending baggage, and eventually we were cleared for takeoff, about 2 hours late.
We have built an overnight layover at Heathrow (with a room at the Sofitel in T5) into our flight, so we had no worries about the delay. But there were lots of people on the plane who were going to miss their US- or Canada-bound flights home. So during the whole flight there were announcements giving the passengers’ last names and what flight they had been re-booked to. A few who were heading to SFO got later flights the same evening, but most people were re-booked to the following day.
Frank II gave you good advice—don’t think about compensation at this point. And focus on making that connection: be ready for the security check by having your water bottle empty and all liquids in the proper 3-1-1-bag. Move quickly to security so your boarding pass is scanned within the required time. Then proceed to your gate (visit the restroom near the gate if you need one).
And if the flight does leave Paris late, listen carefully for announcements as to how to proceed when you get to Heathrow. There will be lots of people on the plane in the same situation and they will address it.
Since this flight was booked with your FF miles, you should be able to change it. Consider calling AA and asking if you can get a better, less stressful routing to Austin. That is how we got the flight with an overnight layover at LHR—-it was booked with Alaska miles and I was able to change the original routing by calling them. I looked on BA.com at possible routes and picked one I liked, then called and asked for that.
