I am looking for a simple mod that keeps time flowing while you're reading to keep the game immersive.
Does anybody know of one?

Okay, this mod goes way beyond what you are asking for but Kuertee's Professions http://skyrim.nexusmods.com/mods/18258/?

adds time to allot of stuff you do in skyrim; like smelting, skinning, eating, reading and the list goes on.

I'm wanting to read in game books, get the irl benefit from them (reading) and still have time pass in game. Will this mod work like that?

Yes.