No, they won't let you use that on the aircraft primarily because it is not 5 point harness; their rules are strict. Have you measured how wide it is and is it in two parts (they do not allow those either). Will it actually fit the BA seat (45x45 cms)?.
It may well be problematic to carry that down that narrow aisle with both your child and hand luggage.
The picture you show is certainly not a BA plane.
At Heathrow as well, they still use buses on some longhaul flights to take people to the aircraft upon which they walk up the steps. If you are fortunate enough to avoid that you will have to use the transit train to get to either B or C gates. It can be hard enough with just hand luggage and children, the thought of carrying a car seat as well makes me cringe.
They will more likely than not make you put this in the hold at check in so it will need to be packaged up with care.
This is also from their website:-
"Car seats
If you decide to purchase a separate seat for your infant for a child fare, or you're travelling with a child who is less than three years old, you may prefer to use your own forward facing car seat.
The car seat:
•can only be used for infants over six months old and children up to three years old
•must be designed to be secured by means of a normal aircraft single lap strap and face the same direction as the passenger seat on which it is positioned
•must have a 5-point restraining harness
•must not exceed the dimensions of the aircraft seat, and should fit into an area of 45cm x 45cm (17.5ins x 17.5ins)
Please note: Two part car seats, (a seat and separate base) are not suitable for use on aircraft seats"
Some other suggestions:-
Do not also rely on them to fully provide re foodstuffs for your child. Take what he likes and enough of it for the flight duration.
If you do have something like a tablet device, I would put lots of childrens programmes on it. I suggest that also because their own childrens tv channel can be quite limited. A list of the films being shown on your flight next month should already be on their website.