Cash compensation for delays under EC261 is entirely separate from the duty of care. Duty of care is mandatory and not dependant on the reason for the delay. BA should pay the cost of your transfer to LGW (or the cost of your overnight hotel at LHR, plus transport to and from it, if you choose that option).

Cash compensation for the delay might be more difficult - protestors breaking into a UK airport and chaining themselves together on the runway is not exactly an everyday occurrence, so BA might well claim "Extraordinary Circumstances".