interesitng WW-- my understanding about cotton was that is was native to china (once saw an interesting print-- made in england that featured "cotton bushes"-- they were envisioned as "mutton trees" --little sheep caught in a bush!)

cotton might not have been grown in ancient egypt, but trade existed... small amounts of silk fibers have been found in tombs, as well as cotton. trade caravans existed , and trade goods from china made it to the near east and europe from very early on.
and since cotton seeds are so hard to remove from the raw fiber, i suspect some seeds ended up getting transported to mid east-- so there is no telling when cotton first started to grow in the area.

and like many wild plants, the varieties we see today, are the most commercial viable.. there might well be some form of periennial cotton that grows in small trees.. but i suspect its it not the most productive variety.