Pegging Westerosi technology at mid 14th century – early 15th century depending on location (mail and leather in some places, full plate and armored horses in others) but with a total lack of gun powder. Assuming gunpowder never appears where does military technology and tactics go in the next 100 years? Woud a Renaissance with out gunpodwer resurect roman tactics? Also assuming the others don’t knock them back to the dawn age

That’s a great question! Without gunpowder, you’re not going to get the same movement from medieval armies focused around knightly charges to the pike-and-shotte tactics of the Early modern era, so you’re going to see a different development pattern. 

In the short term, I think the Golden Company’s model of combined arms and disciplined infantry is going to become dominant once someone realizes how much there is to gain from upgrading the quality of Westerosi infantry and having them work in concert with the cavalry rather than independently. 

You might see cavalry tactics shift somewhat if the Dothraki have a big impact on Westeros, similar to how the Byzantine cataphracts adopted the horsebow and hit-and-run tactics from the Huns, the stirrup from the Avars, etc. Westerosi knights are pretty damn effective already – if you could train them to use composite bows, you’d have a really frighteningly effective force on the battlefield. 

And eventually, someone’s going to figure out how to make an effective repeating crossbow with a decent range and penetrating power, and then you’re likely to see a different kind of pike-and-shotte tactics: