I can sympathise with OP. I experience this regularly - two hours in line for immigration, interrogated by surly and nearly rude immigration officer, queue for customs snaking through bagge hall, transfer to anoter terminal, security staffed by downright rude, hectoring officials shouting at passengers like they are morons........
Only thing is, this isn't at LHR but at ORD.
I'm no LHR apologist. It isn't perfect but it's no worse than most. I would expect, admittedly as a UK passport holder, to be able to do plane door from AA TATL to BA lounge (including the T3-T5 transfer) in less than an hour (it was 40 minutes on Wednesday).