Same experience as devonhub - though with my iPod Touch.
I've synced through iTunes after making sure the audios are in my Audiobooks library in iTunes, then either drag and drop or right-click "add to device" or Sync Selected in the general sync instructions once iPod Touch connected (and btw, have to re-select "manually manage music and videos" every time with iPod in Summary - keeps changing back with latest 12.1.1 update). This always worked before.
Now, it looks as if the audiobooks are "On My Device" before I disconnect, and indeed, icons show up in the Library, but when I click/tap on them on the iPod in an attempt to listen - crickets! Then when I tap the 3 dots, lower right on icon, hoping for a PLAY NOW, only options offered are Delete, remove from want to read, add to collection, mark as finished or share audiobook (and just how would THAT happen??)
So, I add to both Want To Read and My Books, hoping this is magic step. Go to Want to Read and yes, there is the Listen button; again, tapping that - crickets. When I go to same book in My Books and tap, again nothing. Each place holder has those cute little 3 dots that look as if they'll have all the options, but PLAY or LISTEN or PLAY NOW is never there.
I HATE the upgrade 12.1.1 if that's what causing this.