the prisoners receive no information other than the color of their fellow prisoners' hats
All at the same time. Solving the case with guessing in order is a good intermediate step.
With guessing in order, I observe that for every finite subset there are either an even or an odd number of red hats, and prisoner 1 can indicate which it is by his guess; then everyone in that subset can count the red hats and figure out which colour his own must be to make the total number even or odd. Let the size of the finite subset go to infinity.
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.