I think a lot of it depends on whether the pupil is new to fishing or just new to fly fishing.
An experienced fisherman could probably pick up a fly rod for the first time and, with some basic instruction, begin fishing and hopefully catching fish on the fly rod the same day or even the same hour. Of course, if they focus on the fishing rather than the casting then they will probably develop some bad habits and may get frustrated when they wrap the line around themselves or have to play cat's cradle when they are trying to catch fish.
Someone who has seldom casted any type of fishing rod and has little knowledge of fishing would probably be better off getting some casting experience before trying to deal with the "fishing" part, especially if it is a more specialized type of fishing such as fishing in the surf or on the flats or anyplace where they may encounter more experienced fishermen who get upset when a newbie doesn't know "the routine". Of course, if the beginner has a knowledgeable instructor who is teaching him to cast, then the instructor should also spend some time explaining the "fishing" part so that the beginner can learn how to approach a pool or work a river or wade the flats without spooking all the fish or violating the rules of etiquette that are (hopefully) followed by the more knowledgeable anglers. The beginner might not "get it" right away but they will at least have a clue, and as they improve their skills and spend more time on the water they will begin to understand what the instructor was trying to teach them.
p.s. I voted "As many sessions as necessary, then fishing." That could mean fishing on the first day or not, depending on the type of fishing and the skill level of the client.