Unfortunately you are almost certainly due nothing.
Under normal circumstances, you would be due an amount of between 250 and 600 Euro under the “compensation” section of EU261, however the airline will almost certainly claim that the cancellation was caused by “extraordinary circumstances”, which absolves them of paying compensation.
“Extraordinary circumstances” does NOT absolve them of the “right to care” provisions of the legislation. That section states :
Right to care 1. Where reference is made to this Article, passengers shall be offered free of charge: (a) meals and refreshments in a reasonable relation to the waiting time; (b) hotel accommodation in cases - where a stay of one or more nights becomes necessary, or - where a stay additional to that intended by the passenger becomes necessary;
The problem in this circumstances is that no “stay” is necessary. If you agree to the earlier flight, then you will arrive at your destination (as far as BA is concerned) 1 day earlier than planned. It is not BA’s concern that you have a connecting flight the (now) following day – once they have delivered you to your destination (LHR) their commitment is completed.
If you had purchased the two tickets as a single itinerary, then you WOULD be due a hotel in London, as in this case the commitment of British Airlines would be to get you to your final destination, not just to London (this would be the case regardless of which airline the 2nd flight was book on, as long as the flights had been booked as a single ticket/PNR).
Despite no hotel or other compensation being due it would not hurt to at least ask for one. It is certainly possible that BA will provide you something as a customer service gesture, even though it’s not required by EU261.
(And before someone points it out, although the UK is no longer a member of the EU, part of the legislation around them exiting means that the legislation behind EU261 is still in effect and enforceable in the UK for a period of time)