The Peshitta is very much like the Vulgate is in Latin - the end product of several translations of the original Greek and revisions of those translations. The traditional colophons at the end of the Gospels in the NT text in the Bible Societies' Peshitta indicate that the Gospel was preached in other languages than Greek by Matthew and Mark. For instance, at the end of Matthew, we read "The end of the Holy Gospel, the proclamation of Mattai (=Matthew) the Apostle, who spoke in Hebrew (= 'ebrait, "in the Hebrew manner"> in Palestine." Similarly at the end of Mark "... who spoke in Latin (= rhomait, "in the Roman manner") in Rome." Whether that implies the original languages of the written text to have been Hebrew and Latin, respectively, I'm not sure.
Regarding the Diatessaron, it is much more than just a harmony of the Gospels - it was the liturgical text of the Gospels in use in most of the Syriac-speaking world before the Peshitta gained widespread acceptance. St. Ephrem the Syrian even wrote a commentary on it (a source for much of the Syriac text).