I can give the USAF answer.
First, the USAF doesn't call most unmanned airplanes "drones". A target aircraft used for weapons testing might be a drone, but combat aircraft like the MQ-9 Reaper and MQ-1 Predator are "RPA", remotely piloted aircraft. "UAV" is also widely but unofficially used since the "unmanned" part implies that the aircraft is off doing its thing without human guidance, which is not true.
As for RPA time counting for flight pay, yes it does. The USAF considers flying RPAs to be the same regarding flight time as regular flying. The primary difference that an RPA pilot would see when flying combat sorties is that a manned aircraft pilot flying over the combat zone will log "combat" time, while an RPA pilot will log "combat support" time, to make sure that combat time only counts time flown over a combat area.
I am not sure how the FAA considers RPA/UAV/Drone time, however I don't think it can count towards ratings like ATP, instructor, commercial, etc. I think that the airlines probably unofficially consider RPA time to be better than sitting around doing no flying at all, however the last time I looked the airlines used only manned aircraft time when determining application eligibility. Once the FAA gets around to releasing rules for large unmanned aircraft sharing national airspace and people start actually flying unmanned aircraft in the same airspace as civilian aircraft, then it will probably count towards something. However I don't think unmanned flight time will ever be considered the same as manned flight time. There are simply some things you can't learn or teach unless you're actually in an aircraft in flight because the total environment plays a bigger role than many people want to admit. A new guy may be able to shoot an ILS approach in the simulator even in "bad weather" settings in the sim, but put him in an actual aircraft that is bumping around with ATC screwing up and trying to get you killed and an emergency or a flight attendant shouting about something or other going on in the cabin, and its a completely different story. You can't simulate the stuff that happens in the real world so there is no substitute for real world flight time. Just look at that short landing in San Francisco if you want a perfect example of why you MUST have real world experience to be a safe pilot. Those 3 pilots just let the airplane fly right into the ground because their habits and "air sense" were almost entirely learned in the simulator. So when one tiny little thing went wrong (autopilot in an unexpected mode), they didn't recognize it until they were dragging the tail in the water 1/4 mile short of the runway.
So for now, RPA time counts for flight pay and flight status in the military, but it isn't really counted in the civilian world except maybe for sympathy points.
