Both will faint after they moved on the third turn after the move is used. Their speed seems to determine who faints first.
After they both move on the last turn the faster Pokemon will go off the screen first, though it doesn't really affect anything since they both moved anyway.
Source: Replay from showdown with a faster Pokemon, replay from showdown with a slower Pokemon
Hope I helped!
Special thanks to Sciz for helping me test this. :]