Why would need an RTC?
Maybe not because I guess you want to use LDRs to track the sun.
But how do you attack a metal nut to a piece of wood? I could see me screwing a piece of metal to it and maybe welding it? But then again it has to pivot right
Yes it would almost certainly require some welding if you want to do it on the cheap. Not hard but does need some metal fab skills.
You could probably bodge something up, for example a hose clamp can be used to hold the nut, the clamp could be bolted to a plate that is screwed to the wood. The bolt is your pivot point. A search through a junk yard may give inspiration.
I suppose I could get an electric motor that has a threaded drive shaft off of it
I doubt you will find a motor with the rod already attached, normally you couple a motor shaft to the rod.
There's quite a lot involved, even for a "simple" application like this.