Well, they would certainly not have known of Gilgamesh or Enuma Elish, at least not directly, since those texts were quite throughly lost from about when people stopped writing in cuneiform(about 100 BCE-75 CE-ish) to roughly the late 19th century when they were rediscovered thanks to the excavations of Cuneiform archives in Mesopotamia (initially at Nineveh, now on the outskirts of Mosul, later on at a wider range of sites like Nimrud in the north and Babylon and Sippar to the south) and the decipherment of cuneiform giving rise to the discipline of assyriology. Now this isn't to say that aspects of this story didn't survive in other forms or that other myths from the Ancient Near East didn't likewise survive in other forms(and one body of these texts, the Hebrew bible, was after all religious canon!), but the texts themselves would have been forgotten. Most Sanskrit literature like the Baghavad-Gita would likewise have probably been unknown until the late 17th and 18th century with the beginning of Sanskrit studies and really of Middle Eastern and Asian philology in the west as a discipline(this was also the time, remember, when the first efforts were made to study early forms of Iranian like Avestan and Pahlavi); while there is a fascinating story to be told of the foundation of these disciplines and the reception of these texts in the 18th century, I will leave that to others more familiar with the early history of these disciplines.

Now one very remarkable case of a shared literary text(but not an eastern epic per se) that spread throughout Europe and the Middle East is the so-callled Alexander Romance. This text began life as a Greek novel sometime before the fourth century in probably the Eastern part of the Roman Empire and mixes historical literature about Alexander with a wide range of myths about him rooted in Mesopotamian and Classical mythology. Now this story is probably rather obscure to us today but in the middle ages it was astoundingly popular-there were Latin and Old French and Norman translations, but there were also translations of it into virtually every Middle Eastern language in existence and it substantially influenced the portrayal of Alexander both in the West and in the East-for instance, it provided the model for the treatment of Alexander in Persian epic poetry. So it's not quite an "Eastern Epic"(which in any case is somewhat of a nonsensical category that encompasses too many disparate things to be of any real use) but it's certainly a text that would have been familiar both in the medieval west(which I am taking to be more or less Latin Christendom for your purposes) and in the Middle East.