What do these little flying drones on the Uprising map do?
These drones are just for show, that is it. You can shoot them down, but they are only there for lore purposes. They are supposed to just simply be drones on the enemy team. They do not cause damage or contribute at all to the enemy team.
These drones don't appear to be useful for anything. I played many rounds of this new game mode last night, and shot these drones down every time I saw them (even while our payload was being attacked, much to the displeasure of my teammates). I noticed that the rubble that they leave behind doesn't appear to despawn, which is a little unusual for destructible terrain. Other than that, I did not see any score changes or any other easter eggs related to shooting down these drones.