Carry on bags even on someone like BA have been getting crazier and crazier over the last few years. Economy pax now think nothing of taking a trundle-bag which won't fit in the overhead, and another bag, with them.
Thankfully I have FF status which lets me board first so I get my (one, modestly sized I might add) bag into the overhead.
Never book rows 1-3 on BA domestic and turn up late - you won't have any space to put a bag up there - not a chance.
On the 10 or so BA long-haul flights I've done in the last few months, without exception, economy pax baggage has been brought into Business overhead bins as Eco overheads are totally packed. Some of them get on with huge items which I'm amazed they've got past check-in and then security!
If the EU mandate more than one bag, you can forget having any space overhead, ever!
It might be convenient as to put everything in carry on, and as long as it fits in the box outside check-in, fair play, but there are people totally abusing the restrictions already.