I agree with others here - I don't need my hand held. I've transferred through LHR enough times to know my way around.
However, I do agree that travel on BA is largely let down by the experience at LHR.
Providing a genuine transfer experience would make things infinitely better and probably at relatively low cost. For example, a genuine Fast-track transfer, with genuine dedicated security. It really shouldn't be that difficult to organize - we all have boarding passes after all - and they're all scanned on the way into the system.
Also a return to the good old days, where passengers on short connexions were escorted through the system personally.