To complete the details:
I was on BA295, 20/3/16. My original schedule was:
LHR 11:40am - ORD 3:30pm (BA295)
ORD 5:05pm - MSY 7:24pm (BA4449, operated by AA)
OK, so BA295 left the gate at 11:48 (source Flightaware), took off at 12:15 (multiple sources), ORD touchdown at 15:27 (Flightaware), doors open at
16:05 hrs (Expertflyer). This ties in with your recollection.
So you had a small delay at LHR and another at ORD. You also had a tight connection, 8 minutes of the delay (at least) was due to LHR ATC, and you seem to have delays in US immigration as well. So if you had missed the flight by about 8 to 20 minutes - depending how you want to argue this - I think you're probably going to be out of luck here, that 8 to 20 minutes isn't going to be down to BA in my view.
You would need to prove that due to circumstances under BA's control, they could have got you to ORD about 30 minutes earlier and I think you will struggle. [To be more precise, it's actually up to BA rather than you to prove that extraordinary circumstances were involved, but in the case of two of the world's most congested airports with a terminal change and US immigration this doesn't strike me as very difficult].
How did you find out you had missed the flight? At the departing gate? Or beforehand at check-in? Your original service left MSY at 17:05, so you needed to be at the AA gate probably 16:45 to have made that service.
Quite apart from the compensation issue, I'm surprised they didn't rebook you on to the Charlotte service at 18:30, which would have got you to MSY that evening at least. Maybe it was full.
Now that is just my view, and there are other readings, but it's not a cut and dried case in your favour, only a judge could decide.