So you will be landing in LHR Terminal 3 and then go by airside bus to LHR T5. The minimum connection time is 90 minutes, you are way over that (and it can be done in 30 minutes). So you have plenty of time, and your bags checked in in Reagan will go all the way through to Paris. I wouldn't be at all concerned about this connection. In BA Forum Dashboard there is a guide to connecting at LHR, and a similar one in the AA forum, which will give you step by step guidance. But all you need to do on arriving in T3 is to follow the purple signs for Flight Connections to T5. If something went wrong and you were massively late, first check your BA App, to see about whether you are auto-rebooked, but still go to T5 where they can rebook you if necessary.

Scrub that, it looks like you're AA all the way via JFK, and in fact you have a huge amount of time for that connection at JFK, you will be walking around T8 a bit but it's all integrated. If something goes wrong AA should be able to rebook you.