In the memoires of Lieutenant Parquin he describes a parade in around 1807.
Usually French horses had long tails, while the British shortened it (Revell Lifeguards).
On this parade his commanding officer had a horse with a shortened tail and now he tried to lengthen it with some other hair to give it the right length.
As Napoleon mustered the unit, the wrong hairs nearly felt down and only hang on a small part on the original one. Must have looked terrible and Parquin could not help himself from grinning. This brought him a not so friendly response from his CO and later some extra shifts
So I would say the French horses had long tails.
In contemporary sources (El Gil for example) the french horses are shown in long tails too, so there is no reason to think they bound it up.
I do have prints showing French horses with bound tails, so perhaps that was just when on the battlefield, or maybe just a mistake by the artist. I guess unfettered but well groomed tails look better on parade.