I suspect 261 might apply; I can never remember whether its 7 or 14 days notice after which the airline has specific responsibilities. But the Regulation intends to allow passengers a re-route option, if their flight is cancelled, so as to arrive close to expected time.
No doubt some-one can clarify if OP's example counts as a cancellation.

Had this been done 15 days out there is no comeback. Just reasonableness, which BA does better than certain other airlines I've used.

But yes. It is truly a nightmare as a leisure traveller trying to book a family trip on a budget and independently. I am sometimes forced to use separate airlines and can only mitigate OPs type of risk by using flight combos which withstand most disruption (ie. try and avoid last flights, or once a day (or week) destinations). RIX-USH was particularly fun, and we did stay in B.A. on the return. 3 of last 15 holiday trips have had some kind of canc / missed conn on different ticket delay; so it is relatively frequent (less so on BA).