Hi all, tons of great info in FAQs & detailed responses. I haven't seen my specific situation. 8 of us (4 adults, 3 seated children, and 1 lap infant - grandparents, parents, children) booked our round-trip tickets via AA.com in a single purchase. We were scheduled to return home (USA) from the UK (visiting family) on May 31. We were booked on this route:

LHR > PHL on AA 6912 (operated by BA 0067) scheduled to fly 1:40 PM to 4:35 PM local times, then connecting to...
PHL > GSO on AA 5763 (operated by Envoy Air) scheduled to fly 7:13 PM to 9:06 PM.

The BA 0067 flight (LHR > PHL) was delayed due to a staff issue ("sick pilot, waiting for a standby pilot to arrive"). It departed at 4:00 PM (2 hr 20 min late) and arrived at 6:46 PM (2 hr 11 min late). We missed our connection in the States.

Onboard, the BA pilot informed everyone with connecting flights that evening that we should go to BA customer service for re-ticketing connections, after going through passport control. On the AA app, as we went through passport control, we saw that we were already being re-booked for the next day. Apparently, there were no more flights that evening between PHL and GSO.
Our next day's flight was not until ~4:30 PM. We went to the BA desk, waited in a long line, and were sent to the hotel shuttle stop with a voucher for 2 rooms. BA declined to give us any food/meal allowance or vouchers, to carry us through the 21-22 hours we were in Philly, so we kept receipts.

Are we eligible for (delay) compensation (UK261/EU261) since our BA flight delay (2+ hours, not 3 or 4) caused us to miss our AA connection and we could not get home until the following day? If so, what is an appropriate amount?

The hotel and overall PHL experience were terrible. But that's a separate matter. Thank you for your insights and help!