4 Generations (140ish years), if electricity went down.
Once the people who were around for the Fall are dead, it's mostly over as far as retaining modern-day information goes. If you blast humans off the Grid, all you've got are books. If it takes 30+ years to get computers back into use, even things like CDs and flash drives that survived the apocalypse intact will be basically unreadable. So all you've got are hard copies. The problem is modern-day publishing techniques are made for efficiency, not longevity. I have books over 100 years old that are in far better condition than ones made in the 70s-onwards. Even with proper care and storage (which most books won't have) things will be dicey 50 years on. Arizona might get lucky, since it's not really damp there. But even so most everything will go in the years between "everything is on fire oh god oh god it's the apocalypse" and "we now have somewhat successful farming communities."
But even if you have the books, you need people to read the books. Literacy just isn't that big a deal in a post-apocalypse agricultural world. Sure, the 2nd gen will get taught to read, because the 1st gen expects it. But the crops need to come in and the rad-cows need milking and the 3rd generation's literacy rate tanks. With that drop comes less desire to maintain the records people do have. Sure the Bible/other holy works get copied by diligent priests, and agricultural stuff gets copied for its usefulness. But if you get blasted so far back that the 1st/2nd gen can't get the power working again all the technical stuff falls by the wayside. By the 4th generation reading is for priests and whatever nobility equiv gets set up. And they mostly read Holy Texts and practical works History is a super low priority. The Emberverse (starting with Dies the Fire) by S.M. Sterling does a great job of depicting this decline in historical knowledge. If your civilization is blasted back into the stone age and only hear about history via your grandad, who also tells stories about WILD things like Jedi and Captain America, how much will you actually remember? Tanks and robots and Nukes and lightsabers and F-22s and P-51s and T-65s all get mixed up because hey, they were all bedtime stories your grandpa told you about things that don't exist in your world. You tell those stories to your own kid, but they don't believe half of it. Sure everyone knows a giant gorilla called King Kong lives on a far-away island. But some war that killed 100 million+ people? IMPOSSIBLE! Even with the Ancient Weapons.
So the stories get mixed up, and the books are dissolving away, and only the most successful immediately-post-fall communities managed to keep any non-survival-oriented books at all. But it's been over 100 years and nobody is QUITE sure what the difference is between fiction and nonfiction, and hell the world's changed anyway. Not efficient to spend a ton of brainpower figuring out what's real and what's not. The harvest doesn't care either way!
TL/DR In 4 generations, give or take, pre-apocalypse history will be mostly forgotten, and what is remembered will be so mixed up with pre-apocalypse fiction nobody will have any idea what's true and what's false.