Between leaving the aircraft and getting to the bus, there's probably always enough walking distance and time to fire up your smartphone, connect to the free WIFI and check up-to-date informations regard your next flight.

In any case, transferring between T3 and T5 is a piece of cake. On your way back (LYS-LHR-RDU), you'll might have to pop into the (AA office, located after security at T3). They'll check if you're all set for your flight. In my case, they completed passenger informations.