How do drones overcome latency?
There are multiple ways to fly and control Unmanned Aerial Vehicles (UAV), or drones, for e.g. the Predator.



Source: science.howstuffworks.com


Some UAVs (like the Global Hawk) can perform the entire mission (from takeoff to landing) autonomously, negating the need for a pilot.
Most of the UAVs, like Predator can perform (i.e. fly in) some simple missions (like reconnaissance) autonomously.
In case a geostationary satellite is used for communication with the UAV, the latency is around 300 ms (The electromagnetic waves have to travel ~70000 km). In case of other delays (like equipment etc.) this will not be greater than 500 ms or around half a second. This is not significant in case of slow moving systems like Predator (they are not dog-fighting, after all) and the 'pilot' can effectively control the aircraft.
The only situation where this is a problem is when the UAV is taking off/landing and in this case, the control is handed over to local LoS operators. 


The Predator is reported to have a 'latency' of around two seconds, which causes problems during takeoff/landing. According to the telegraph,


  But the two-second delay between a pilot moving a joystick in Nevada and an aircraft responding in Afghanistan is enough to cause a crash during take-off and landing. Crews in Afghanistan control 'launch and recovery’ through direct contact with antennae on the aircraft. Half an hour after take-off, control of the Reaper is handed to a crew in Nevada; half an hour before landing, it returns to the crews on the ground in Kandahar. 


Note that the RAF also does things the same way. According to the report on formation of 'drone' squadron' at RAF Waddington,


  Although the personnel will be permanently based at RAF Waddington, some will go to Afghanistan to control the take-off and landing of the drones locally.