Fender Jazz Bass has a nut width of 1-1/2" , the Fender Precision can be 1-5/8" wide nut or sometimes 1.65" (slightly wider than 1-5/8")
...and some old Precisions were 1-3/4" nut width.
I think the nut width on the old 1968 Fender Tele Bass neck Billy had on the original "Wife" Bass was a 1-3/4" nut width, but I do not know how fat or not fat (flat) the neck was.
I am from Rochester New York and Billy is from Buffalo New York, so we would see him with his band Talas in Rochester bars in the middle to late 70's (he's five years older than me).
He had the loudest bass rig I had ever heard at the time, my eyes were jittery at the front of the stage the first time I heard him.