OK, so it's pretty clear that it is one piece only ex-LHR. For the benefit of anyone passing by, the
official line from BA is
Passengers departing from or transferring through the UK may only take ONE piece of Hand Baggage
* Due to airport security restrictions currently in place in the UK, only ONE piece of Hand Baggage will be permitted when departing from or transferring through the UK.
So why does my ex-LHR BP say
Hand baggage allowance
1 bag+laptop/briefcase
I think I'll take my laptop bag out of my carry-on, and try to go through security showing my BP with this info. Crazy!