Having flown with BA more than once, in Business and in Economy, this is my experience:
What @Willeke mentions regarding workers' safety actually refers to 32kg. They won't go over 32kg, not even by half a kilo.
On paper, as @German says, the 23kg limit is per piece of luggage. However, de facto they can be more flexible. Personal experience: once I showed up with a 27kg suitcase, when I was allowed one 23kg piece. I offered to take something out (I'm a book collector, so I could have one big tome as a "personal item"), but they just waved me on and said "don't do it again". The measure of their flexibility might depend on how full the flight is (the flight was half-empty), and/or whoever is sitting there. However the likelihood of flexibility rather increases if your total is within what you're allowed.
The 23 kilo limit is to save workers from damaging their back, so they usually keep it strong.
As you do have two cases, re-arrange your belongings to neither one is over 22 kilos (leaving a bit of space in case one or more of the scales are off.)
There are ticket classes which do allow for 32 kilo luggage. Which is over the 23 kilo range by quite a bit, but those cases are handled in a different way and airline staff are required to handle them with two people.
Strictly speaking, it's per case. In my experience they'll often be a little lax if one is over by a few hundred grams. I would not expect them to accept a case that is four kilos over the limit, so I would move some items from the heavier case to the lighter one before checking them in.
Another option would be to put the heavier case on the scale first to see whether the agent at the desk says anything. If they don't, you don't have to do anything. If they do, you can transfer the items at that point.
My own experience (not specifically with BA, but more generally):
The hard limit per piece in many countries is 32 kg, to protect workers. In countries where it applies, you definitely won't get anything beyond that weight. Either you move stuff to other bags, or you'll have to send the bag as freight.
Under that, most airlines will allow you to split the total allowance on the booking (which may be for several people) as you want. You have a max number of pieces, and a max total weight, and you do whatever you want with that, they just add up the weights of the bags and count them. Some airlines are explicit about it (though not necessarily everywhere), others a lot more vague. Note that this may also vary based on the origin/transit/departure countries.
For airlines which don't allow merging the allowances, you can always pay for an overweight bag. It's usually cheaper to do it online before the flight.
Storytime: When I was a teen, I worked for a moving company. When a box was over 50lbs, (~22.6Kg for you Europeans!), there was an exorbitant fee. This fee, supposedly, went directly to an insurance policy for the employees. The idea was that if we are lifting heavy boxes at risk of injury, then at least we would be protected if our backs gave out.
This policy is likely similar or identical to the policy of the airlines. Airline employees have to lift heavy items all day, and a generally agreed upon limit for heavy items is roughly 50lbs/23kg, as determined by medical professionals. Airlines could be at risk of union or employee retribution if they forced employees to engage in activities with serious health risks.
the 23kg limit is per luggage. So, if you have two luggages it is correct you have 46 kg but must split you weight in both cases. You should move some items to the lighter one. Otherwise you will have to pay for extra weight (even if in total you have less thank 46kg)