Well, first, the OP says “full scale nuclear war”. Under those circumstances, many prisons, being more removed from major population centers in general, are going to be the lucky/unlucky ones depending on whether you think surviving or dying quickly is “lucky”.
I figure in the immediate aftermath, everyone is going to be looking for themselves, their families, and trying to find a place of safety. A prison -might- look good for the last, but the power grid is going to be down (forever) and outside of a very few areas where the climate and supplies of local water are just right, are utterly and totally unsustainable.
Whether or not there is a power-free lockdown option for the cells (once any emergency on-site power is exhausted) means that there may be a short term moment of freedom for the prisoners, possibly with existing cliques sharing power, trying to find supplies or safety, or just freaking anarchy and escapes, it’ll end pretty fast as it will for most survivors - no real long term supplies, no likely probability of growing food (due to toxic soil from fallout or nuclear winter), certainly no effort to rescue such by any remaining military/governmental successors as being a very low priority in terms of continuation of order/state.
And then they die. And you die. And possibly the species dies.
Now, the question is if our OP @Cmyers1980 is just curious about the short-term between the immediate exchange and the death of most living persons on the planet, the rest of the thread hits most of those option, IE neglect, benign or otherwise followed by escapes or death. Or if the OP wants to consider the scenario of a limited exchange, say a few dozen to the low hundreds of nuclear weapons, that might be different. In the the limited case, with martial law and a general breakdown of more civilized society, I could see prisons repurposed to secure facilities and prisoners becoming literal chain gangs of forced labor being worked to death as ‘expendable’ (at least Max and Supermax lifer prisons).
But that’s not the OP as written.
Enjoy your Halloween nightmare fuel!