For departing from Dublin just bring your drivers licence as photo ID is sufficient ... ID will be checked by the boarding staff, but drivers licence is fine ... There is a desk at Heathrow when you arrive but in the hundreds of times I've flown there I've never seen anyone actually at the desk and even if there was again a drivers licence is sufficient ID.
On the return from LHR there's no check except sometimes at the gate - in the past BA never used to check ID but recently they have sometimes, but again a drivers licence is just fine .. there's no other ID check at LHR
I flew from Cork to Heathrow in September (aerlingus) with just a driving licence (passport was out of date). I had no problems either end.
I was actually very impressed with heathrow staff in general, there seems to be a change in the attitudes of staff and I found the whole process less stressful then before.
Before the scanners as others have pointed out they only want your boarding card. The last time I was there they sent someone who was stressed and in a rush back to check in because the bar code was not working on their paper print out. Also noticed that now you can scan it on your laptop etc. No idea what one does if the PC breaks down, but I always keep a spare paper boarding card in my bag.
Good tip to have spare boarding card. The shops in the dutyfree area look to see it. I lost mine this way once. No fun trying to get on board flight without it. I was not allowed to return to place where I had left it. I also save a digital copy to device that I travel with. This is useful to do first in case printer malfunctions.