I flew on cx out of ord a few month ago (came in on a separate aa flight on a separate ticket) - and took the shuttle bus (I had only carry on luggage). I was able to check in on Cathays website 48 hours prior and then I was able to get a boarding pass at 24 hours (it told me I couldn’t get it at 48 hours when I initially checked in).

My suggestion would be to go to the flagship lounge in t3 as long as you can (appears you are exp) - as the British airways first lounge in T5 is tiny and packed. The folks in the aa lounge know the schedule of the bus - and it was down for about an hour around 1 (I had to go over earlier than I wanted).