If you had not run the tractor at all before draining the oil, you might have gotten away with one drain, filter change, and refill. Water will tend to settle on the bottom of the reservoir and should have been the first thing to drain out.
Since you operated the tractor enough that you see milky fluid, you have to consider that you have contaminated the entire system: 3PH, FEL, filters, and HST/filter if it has one. This is a much, much more difficult task. You will certainly want to change filters, refill, operate, drain, change filters again and refill as a minimum. If you can find some cheap oil that meets JD specs, use that for the first refill. You also can probably not completely refill and be okay if the tractor is on level ground. For example, if you have an 8-gallon sump, you will probably be okay with 6 gallons, maybe even 5 for the first flush. You should not have to completely refill until the second refill and filter change.
I'm sure glad you didn't submerge your entire tractor and yourself. Everyone of us who operates around ponds has to keep in mind where we are and how narrow our safety margin is. I mow by backing my tractor down the pond dam toward the water, and I always make sure I have plenty of fuel in the tank. If my engine were to become fuel starved and die, I'd find myself in exactly the same situation as you. It can easily happen to anyone of us.