A Canticle for Leibowitz is set in the past of the Dune Universe.
Dune is set ~10,000 years in the future. Canticle takes up 1,200 of that. The "Butlerian Jihad", in which humans got rid of computers sounds a lot like The Simplification. While the two were not the same event, the history of Canticle likes repeating itself.
- Oddly enough, this makes sense...as the humans in Dune aren't aware of where Earth is anymore (possibly because the knowledge was lost/deliberately suppressed in order to give humanity a fresh start) and the Bene Gesserit, while being an exclusively female religious order, are run much like the Jesuits (possibly because their origins lie with the original church-sponsored colony ship).
The Book of Eli is set in the past of A Canticle for Leibowitz.
It takes place before or during The Simplification. Either Alcatraz was eventually destroyed, or it managed to stay secret until after the first and second parts of the book.
A Canticle for Leibowitz is set in the past of the Simoun universe.
See the Simoun WMG page for details.
The world of A Canticle for Leibowitz is not our future. It's our distant past.
The world of the book is simply too different from ours. There are immortal beings and mutated zombies, and what appear to be psychic powers. When the second nuclear war destroyed the world, the ship of refugees that was sent out to preserve mankind landed on a conveniently habitable planet which was then named after Earth-That-Was. Thanks to Eternal Recurrence, our present civilization ended up taking on more than a few features of theirs, minus the Medieval Stasis. Of course, history has become so muddled over time that our origins have been long since forgotten, but the story was preserved in a chronicle of the events by an obscure author who never wrote anything else.
Oh, and Benjamin? He's dead. Atomic blast hit him. Nothing could be done.