I wonder if they OP meant they would be arriving into T4 (main) - which is (as a terminal area) Schengen cleared.

Despite the UK not being in Schengen, the IB operated flights to LHR depart from T4 main (with schengen checks done at the gate).

The BA operated flights depart/arrive at T4(S) - the sattelite terminal, and the non-schengen upstairs level of it.

To complicate matters, as oneworld partners they code share on each others flights, so you need to work out whos metal you are flying.

rdodger: If you will be on one ticket, they will not sell it to you if it is below the MCT. Additionally if on one ticket, they will sort you out if you, and/or your luggage mis-connect.

If you are on separate tickets: They will only accept your luggage for through checking if it would be compatible with the MCT. If you are tighter than the MCT they will not accept it.
Additionally, if you are on separate tickets they have no obligation to look after you, rebook you or anything else if you don't connect due to minor delays. While they MIGHT well attempt to rebook you, if it is busy that could be days later, and they simply might not do it. And they almost certainly won't accommodate you if necessary overnight etc. Further more, most travel insurance requires people to have allowed a sufficient excess connection time to any 'unofficial connections' before it will cover them.

Generally as an answer, 45 mins is WAY WAY too tight.
Even if you were not wanting to put bags in.

You could land on time, and not be on stand for 5 mins.
They will then take the time to off load the bags (containerised), open the containers, have the system send the bags to the onward flight.
Have them load the containers with bags, then load the aircraft with containers. They will probably want to have begun boarding at least 20 mins before departure.

For an easier trip, get yourself onto a BA flight LHR-MAD (even booked under the IB code, but operated by BA), you will come into the satellite terminal where you next flight departs.
You will go through security, but not enter Schengen (only to leave it again moments later).

But, on separate tickets, I would NOT try 45mins, even without bags.
I might just test it going from the longhaul TO the shorthall without bags, but even then I would be worried.