From regulation 261/2004 (which is relevant even though flights are next summer)
Article 5
Cancellation
1. In case of cancellation of a flight, the passengers concerned shall:
(a) be offered assistance by the operating air carrier in accordance with Article 8;
(b) be offered assistance by the operating air carrier in accordance with Article 9(1)(a) and 9(2), as well as, in event of re-routing when the reasonably expected time of departure of the new flight is at least the day after the departure as it was planned for the cancelled flight, the assistance specified in Article 9(1)(b) and 9(1)(c); and
[...]
Article 8
Right to reimbursement or re-routing
1. Where reference is made to this Article, passengers shall be offered the choice between:
[...]
(b) re-routing, under comparable transport conditions, to their final destination at the earliest opportunity; or
(c) re-routing, under comparable transport conditions, to their final destination at a later date at the passenger's convenience, subject to availability of seats.
[...]
3. When, in the case where a town, city or region is served by several airports, an operating air carrier offers a passenger a flight to an airport alternative to that for which the booking was made, the operating air carrier shall bear the cost of transferring the passenger from that alternative airport either to that for which the booking was made, or to another close-by destination agreed with the passenger.
[...]
Article 9
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;
(c) transport between the airport and place of accommodation (hotel or other).
The regulation cannot compel BA to book you to LHR, but its certainly worth a shot asking the agents.
Given the time between now and the flight its highly unlikely that BA will arrange a hotel for you, however from reading the regulation it is their obligation. Use this as leverage to get on the LHR flight saying it'll cost them more to accommodate you than it would to transfer from LHR to LGW.