Jane --
I'm not a family researcher, but will do what I can. You have probably noted the list of Treadways that David Wiggins provided here --
http://history-sites.com/mb/cw/gacwmb/index.cgi?read=6518
The J. D. Treadaway who appears on both 14th Georgia Artillery Battalion and King's Georgia Battery is probably the same man.
You mentioned Joel D. and Joseph Treadaway. The only service file(s) that use this name are for a man who served in Lumsden's Alabama Battery (Co. "F", 2nd Ala. Artillery Battn.), a company recruited in Tuscaloosa, Ala. Both names appear with this unit. Is this the right man?
What do you know for sure about Joel D. Treadaway? For instance, do you know where he lived in 1860?