I have a couple different versions of NudeByte downloaded and unpackaged, but I'd like to get rid of the older ones so I don't have them taking up space. My concern is, how is my save file stored? I don't want to go accidentally deleting it.
It's safe to delete older versions of Nude Byte. This will not affect your save data.
To give a technical explanation: the save data is stored in your Windows' Registry. This means your save data is stored inside Windows itself. This also means it's never saved in the game folders. So, regardless of the versions or file location of the game's folder, the game can always read your save data.
(The only time it can't read your save data is when you play it on a new computer. If this is the case, you will need to transfer your save data from one computer to another. There's a "Save Transfer" software in the list of Downloads above for this.)