I doubt if anything but minor flesh wounds would be treated. If the horse was badly lamed or unable to eat it would be shot: I think that would have applied through to the middle of the twentieth century - speaking here as a veterinary surgeon.
An example of a treatable wound is in Macdonald's 'Souvenirs': at the end of the battle of Wagram, "I then noticed for the first time that my horse had received a bullet in the neck, but which had remained between the skin and the flesh; he was taken away in order that it might be extracted.'

Susan