Yes, BA and IB are at the same check-in and departure terminal in MAD. But you'd arrive at T4 Satellite, have to clear inbound immigration (and presumably on UK passports you'd be stuck in a queue with other non-EU citizens coming from Latin America), take the train to T4 main to collect bags and re-check, clear security, take the train back to T4S, clear outbound passport control (again in the non-EU queue), then find your flight.

On a perfect day 1:50 is doable. Anything other than a perfect day and you'd risk missing the BA flight. If you are on EU passports I would be more comfortable with 1:50.

Separate tickets won't be protected, I cannot see a situation where BA would willingly move you in the event of an IB delay (but of course you could always buy a BA ticket that allows FCOD to help you out with that).

Almost no chance of through-check.