Write to BA and say
“You have re-routed me from London City to London Gatwick Aiport. Under article 5 (Cancellation) of EU261/2004, I have the right to assistance. Under Article 8 (Right to Reimbursment or Re-Routing) states “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 trans-
ferring the passenger from that alternative airport either to that
for which the booking was made, or to another close-by desti-
nation agreed with the passenger”
I propose that you will pay for a taxi between my address and Gatwick Airport, in both directions.
They will probably not agree to this, but you may get a return rail fare.