Good point. I believe the effects on linguistics would also depend on who controlled the means towards literacy.
Say if it was a decentralised movement, whilst we would not see many changes in the dialects themselves (or at least less drastic ones), we would probably see more dialects develop into actual widely practiced languages.
In many countries at the time of the POD there were in fact many different dialects prevalent in the same nation, it was only through centralised literacy programs that one language really came to dominate.
If literacy is achieved in a decentralised way, then I could easially forsee countries (I'll use France as an example) with huge linguistic differences in their borders, as was in fact the case for most of history.
It depends also of the kind of writing you are using. Most of non-european countries prior 1800 or so had literacy based upon writings that are not entirely alphabetic: logographic systems, abjads, consonant-based abugidas, or varying mixing of the above (like Japanese). Even alphabet-based sysetmes may shift towards a writing standard removed for actual pronouciation, as it is the case for modern French, English and, AFAIK, Gaelic.
Most of such systems are more conservative and, OTOH, more difficult to learn than phonetically-based systems, but are also more free from dialectal influx, or, better said, they are less prone to feel pressure from below.
Arabic has a great variety of dialects, but the nature of its writing systems makes differences far less apparent in a written form, even if some are still there to be read.
The problem with a dialect-based widespread literacy is that there may be not that much worth reading in a single dialectal area of mutual understandability, since cultural production will likely still be an elite thing.
Literate farmers would still need to be farmers.
Such a literacy would also mean that atext written relatively few hundred miles away would unavailable to the majority.
Dante's Comedy would be understood only in Tuscany and neighboring areas, Shakespeare might write in a Stratfordian vernacular hardy intelligible in London, nobody in Lyon would understand Rabelais. Not that Dante, Shakespeare or Rabelais could write in this world what they wrote OTL, actually.