Since there were plans on at least the Soviet side for invading Western Europe, how would've they organized the supply of the Warsaw Pact troops? Would the major ways that which logistics would've been handled have changed a lot as the Cold War moved forwards?
No not especially. WW2 saw in mass the move from draft animals to combustible engine carriers.
Supply lines are along major roads and rail lines. You'll have warehouses in the rear and you have to reestablish or establish new ones as you move forward in a leap frog effect. You also had to look at how spread out or tight together armies were.
Too close together and you have a hard time stockpiling enough supplies for an offensive operation since you're literally feeding an army. If their too spread apart, your transit time increases to resupply troops.
If interested about more, you should read a book called Supplying War. The author talks about Napoleons invasion into Russia, Hitler's invasion into Russia, and I believe the Austrian-Prussian conflicts or Franco Prussian conflicts.
Does that answer your question?