The gatling gun in general, but especially the early models had some serious drawbacks that made them less suitable for the kind of warfare being discussed here. Basically, and as others have mentioned, they fell into an odd category somewhere between infantry and artillery. They were moved on an artillery carriage, which limited the mobility to that of an artillery piece, but the rounds they fired put them into more of an infantry role. So right there you have a problem of how eactly to employ gatling guns. The problem wasn't helped by the fact that the gun needed a fairly high and exposed firing position and had to be relatively close to the front due to its range. This makes it particularly vulnerable to fire by opposing infantry and artillery. Also remember that until the 1880's gatling's could not befired continuously - you had to stop and reload it after firing a stack of cartidges. Later developments fixed that by letting the operators fire from one stack while reloading another. Additionally, my understanding is that the early models were fairly cumbersome to reorient. Now, none of these is a huge probelm, but each is another small issue that chips away at the weapon's effectiveness in formal industrialized warare. There's a reason such a high percentage of their use was on ships, especially early on; the nature of the platform negated a number of the gatling's disadvanatages. This isn't to say it couldn't be an effective tool for European warfare, if utilized properly it absolutely could, but there are enough problems with the gun, and enough ways to neutralize it that the gatling gun could only be employed in a relatively limited manner. It's worth looking at where the gun was employed and to my knowledge it saw its most effective use in colonial conflicts where the mobility was no worse than an artillery piece, but the firing characteristics made it a much better match for the role than a field gun. Colonial campaigns didn't often need field guns to blow open enemy positions or formations, but they did need a way to maximize the firepower a small unit could project.
With regard to the conflicts mentioned:
Austro-Prussian War: I don't see it as likely. In 1866, the US had only just adopted the weapon and it wasn't being manufactured in large quantities. It was also still a problematic mechanism. So I don't think it's even a possibility for this conflict.
Franco-Prussian War: I believe the gatling was in fact purchased by France in the later stage of the war, but even if purchased before the war it wouldn't have been significant. France simply didn't have a good doctrine for using them, and that won't suddenly change if they're using gatlings instead. Additionally, I don't see it as plausible for Prussia to buy gatlings. First of all, Prussia to my knowledge does not have a history of ourchasing American military equipment in the period - France does. Also, and I am by no means an expert, but as I understand the Prussian doctrine, gatlings don't have a valuable role in that.
Russo-Turkish War: This one might have real potential, I think. Both sides used gatlings during the war, but neither made significant or effective use of their weapons. Perhaps if the Ottomans have larger numbers, and use them for stiffening prepared defenses some of their positions might hold out longer?
With regard to the conflicts mentioned:
Austro-Prussian War: I don't see it as likely. In 1866, the US had only just adopted the weapon and it wasn't being manufactured in large quantities. It was also still a problematic mechanism. So I don't think it's even a possibility for this conflict.
Franco-Prussian War: I believe the gatling was in fact purchased by France in the later stage of the war, but even if purchased before the war it wouldn't have been significant. France simply didn't have a good doctrine for using them, and that won't suddenly change if they're using gatlings instead. Additionally, I don't see it as plausible for Prussia to buy gatlings. First of all, Prussia to my knowledge does not have a history of ourchasing American military equipment in the period - France does. Also, and I am by no means an expert, but as I understand the Prussian doctrine, gatlings don't have a valuable role in that.
Russo-Turkish War: This one might have real potential, I think. Both sides used gatlings during the war, but neither made significant or effective use of their weapons. Perhaps if the Ottomans have larger numbers, and use them for stiffening prepared defenses some of their positions might hold out longer?