Sneak attack doesn't trigger off having one of your allies adjacent to the target. It triggers off having an enemy of the target adjacent to the target. If the mount is visibly aggressive towards the target, it should count as an enemy.Does a mount count as an ally? Certainly an intelligent mount like a centaur who is allowing you to ride them is. But a trained warhorse you’re controlling? Harder to say.
They don't lose their reactions, but they don't have free rein (literally) to make use of them, nor do the rules for controlling a mount include the option to direct them to do so. It's putting it too strongly to say that the mount cannot think for itself while controlled, but it certainly can't act on its own behalf.Can a controlled mount make an opportunity attack? I don't remember being told they lose their reactions, so I would tend to rule that they could. That would make them a threat I'd say.
Knowing that the horse may kick you in the balls if you devote too much of your attention to the rider is distracting, though.There is a non-technical part to Sneak Attack: "strike subtly and exploit a foe's distraction." Attacking from horseback is not subtle, and it tends to draw attention, instead of distract.