As Paxboy says, Fast Bag Drop has become a bit of a farce. As to queues, I'm afraid you are in the lap of the gods. Economy can be really horrible but BA staff do try to sort things out and you'll pass that part given a sensible arrival time at the airport.
As to Security, on a bad day and it can be any time of day the queues will build back out of the terminal. I would allow 3 hours to be safe. If fasttracking, then usually no more than 30 minutes for check-in and security. Murphys law of course means you'll swan through in 10 minutes but c'est la vie.