For most of the books of the New Testament, there is no tradition competing with the view that the book was originally written in Greek. I am aware of ancient tradition claiming a non-Greek origin for 3 of the New Testament texts: Matthew, Mark, and Hebrews.
Every early Christian historian who wrote on the matter stated that the Gospel of Matthew was originally written in Hebrew, not Greek. Relevant testimony was given by Papias, Pantaneus, Irenaues, Origen, Epiphanius, Eusebius, Jerome, and others. This site offers a review of the statements made by early church fathers on the matter.
Those rejecting this view must claim that the early historians were unanimously wrong. This is a tall order considering that Pantaneus claimed to have a copy of Matthew in Hebrew, and Origen was one of the greatest Hebrew scholars in Christian history.
A textual witness to a Hebrew original may survive: Shem Tob Matthew. It's a copy of the Gospel of Matthew in Hebrew that was written into a 14th century Jewish polemic against Christianity.
George Howard, who published an extensive review of Shem Tob Matthew (The Hebrew Gospel of Matthew), contended that the Hebrew text was not a translation but was a corrupted descendant of an original Hebrew composition. One of the most compelling pieces of evidence he marshalled was the existence of numerous word-plays in the Hebrew text that are not present in Greek.
Since the author of the 14th century polemic was trying to discredit Christianity, it is implausible that he introduced literary flair into the text of Matthew--that would make the text look better; he wanted to make it look worse. If Judaism had in fact preserved a corrupted version of Hebrew Matthew for ~1300 years, it is unlikely other Jewish scribes would have added literary flourish to the text either.
My own work on the Synoptic Problem has led me to the conclusion that Matthew was originally composed in Hebrew, and the Greek is a translation.
The principal reasons many modern scholars reject the claim that Matthew was originally composed in Hebrew are:
- The belief that Hebrew was a dead language in the 1st century (The Dead Sea Scrolls discredited this claim; I review some of the evidence here: What Languages Did Jesus Speak?)
- The belief that Mark's Gospel was written first (I show that each of the major arguments for Markan Priority is circular, reversible, or both here: Deconstructing Markan Priority)
- The belief in naturalism (which, when applied to the New Testament, is question-begging)
Although early historians are unanimous that Mark was composed in/near Rome, they do not specify whether it was composed in Greek (the lingua franca of the eastern empire and the early Christian church) or in Latin (the lingua franca of the western empire and of Roman government). There would be nothing unusual about a Christian missionary from outside of Rome teaching an audience in Rome in the Greek language: it's the one language both the missionary and everybody in the audience could be expected to understand.
The overwhelming majority of the evidence (manuscript, textual, patristic) supports the view that Mark was composed in Greek. Competing with this view are:
- The fact that Greek Mark has more Latinisms (Latin expressions ported into Greek) than all the other Gospels combined. This could be because it's a translation from Latin, but also could simply speak to the fact that it originated in a place where both Latin & Greek were widely used.
- A single manuscript, Miniscule 13, written in the 13th century, which directly claims that Mark was originally written in Latin.
Most sources accept that the Epistle to the Hebrews was originally written in Greek. Competing with this view is the statement by 2nd/3rd century scholar Clement of Alexandria who stated:
the Epistle to the Hebrews is the work of Paul, and that it was written to the Hebrews in the Hebrew language; but that Luke translated it carefully and published it for the Greeks (as preserved in Eusebius Church History 6.14.2)
The evidence for a non-Greek origin is strongest for Matthew. There is some evidence (later, less consistent) for a non-Greek origin of Mark and Hebrews. I am unware of any ancient witness supporting a non-Greek origin of any of the other 24 New Testament books.
Re chronology, referenced in the OP, for a textual & chronological argument that the Gospels were written earlier (and for the Synoptics, much earlier) than 70-80 years after the crucifixion, see my video series here: Who, When, and Why - the Writing of the Gospels.