Based on a post on WoWhead you do indeed get both the Horde and Alliance mounts from completing the Pre-Expansion quest the first time you complete it.
By joezu (3,433 – 13·26) 17 days ago (Patch 7.3.5)
* Does anyone know if you get the Horde bat too if you complete the scenario on an Alliance character?
Alechs777 16 days ago (Patch 7.3.5)
* I’ve read posts on the forum saying that yes, the first time you complete the scenario you get both mounts.
Reignac 16 days ago (Patch 7.3.5)
* Yes. At least you do on beta, but I bet that will stay on live.