Right to care exists at all points while you are an airline customer, so it's really only at the point that you ask for a refund that it would stop. Rebooking is one of the remedies that applies in the event of cancellation (and in some extended delays). There is not a direct requirement to rebook outside BA if that is quicker, but some caselaw exists that indicates that if BA did not do that then they may have to pay compensation even in extraordinary circumstances. The CAA has some guidelines which indicates their expectation that people should be rebooked the same day, which kind of implies that airlines can expect you to be delayed within the same calendar day but not to the next day if other airlines are running. As it happens BA would have rebooked to TAP via the contact centre had that been staffed to adequate numbers (and there is a separate CAA guideline that says airlines should be able to cope with meltdowns). Also in some cases MMB would give options outside BA too, but not in this case as I understand it.

What it boils down to can be seen via McDonagh v Ryanair Ltd, where the passenger had unpaid expenses when Ryanair cancelled services. The claim against Ryanair was successful for a number of reasons but a key test was whether the passenger's claim was "necessary, appropriate and reasonable". So if staying overnight is unreasonable due to (e.g.) children having to go to school the next day, then the airline cannot insist on an avoidable delay.