Skipness One Echo: T1-T4/T5 transfer
This is what the BA web site says about transferring to T5 (and T4 is no different) from gates 80-90 (ie the Ireland arrival gates) in T1 Domestic.
"You will arrive from Ireland between gates 80 - 90 in Terminal 1. From here you should make your way to the main terminal, following the signs for the exit or baggage reclaim. Once you are in the main terminal you should make your way to Terminal 5 using the Heathrow Express, the Heathrow Express runs between the Heathrow terminals and then onto London Paddington."
The above is in line with what I was told by Aer Lingus after I'd spent over 2 hours transferring between T1-T4 about three years ago. Needless to say, I've never used LHR as a transit airport since! My own experience was made worse because the Heathrow Express wasn't working. Well why would it be? It was only the Easter weekend we were travelling over. Not a busy time or anything, right?