I wont speculate on how both sides might have done together, WAY too many variables and frankly it really isnt the point of this sub. But the Union itself had created an effective army, capable of operations on a continental scale, and a strong understanding of both modern tactical, operational, and strategic skills, and a massive logistical train including extensive railroad usage. And with about 2.25 million men serving in the Union Army alone it would have been comparable to the combined number of soldiers under arms by both France and Prussia in their war. But it took the Union over 3 years to reach those numbers and only about half that would have been serving at any one time, so obviously not only did European nations have much larger regular forces to start but were able to mobilize well trained reserves quickly instead of relying on raw volunteer units of variable lengths of service.

One thing they did not have though was any sort of robust cavalry arm. Really from Washington on down the existing terrain, resources, and exigencies of war meant US interest and esources in cavalry was lukewarm at best. And what units existed were almost universally light cavalry or dragoons used to police the frontier.

During the war outside of a handful of leaders cavalry on both sides was reduced to scouting, and raiding, with little tactical usage, and most of that dismounted. When cavalry did meet cavalry in numbers it quickly became very disorganized such as at Brandy Station, Gettysburg and Yellow Tavern.

European cavalry units, in particular heavy cavalry well versed int he sword, or lancers would have been quite a change. Though conversely so would the Union's late war usage of cavalry under Sheridan as essentially fast light infantry, in particular when they raced ahead of Lee and fought on foot to bring him to a halt at Appomattox while the infantry caught up.

The Union also had an extremely robust river fleet, which few of the European powers had much reason to invest in, besides for coastal defense.