http://www.jwstudies.com/Why_Does_WTS_Accept_Christendoms_Scriptures.pdf
and http://oyc.yale.edu/religious-studies/rlst-152#sessions
also offer insights into what transpired back in the day.
These are some of the thing which I think were significant contributors:
1. There was a maelstrom of conflicting ideas among the various early Christianities. The Paulines (who owe their origin to the Apostle Paul) eventually won out and became the dominant force in Christianity; all other groups were branded as heretics.
The first followers of Jesus were Jews, as Jesus himself was, and these followers saw Jesus as a wise teacher, not as God. It is said that Paul embellished the Jesus story in order to undermine the Jewish Christians so as to take power away from them.
2. Of great importance is the fact that some of Paul's writings pandered to the Roman authorities e.g. Paul's writing in Romans 13 calls for submission to governmental authorities, although passages in 1 Corinthians may be said to contradict this. 2 Thessalonians, a pseudonymous letter, also preaches a politically conservative and accommodative message.
(http://oyc.yale.edu/religious-studies/rlst-152/lecture-24).
Given Paul's pandering to the Roman authorities, Constantine saw the political value of Pauline Christianity - Pauline Christianity was not just seen as a religion but as a political solution to bring about the unity and harmony and control of the populace which Constantine desired.
With this in mind, one can understand why Constantine and Theodosius promoted the Paulines out of all the other conflicting voices of early Christianity.
3. There were only a certain number of writings which were recognized as Christian writings by the Catholic Church. From that number, the books which comprise the NT were selected by the Catholic Church.
The Gospels, Pauline letters and other NT texts which were selected were a midrash on the OT scriptures and as such the OT and Torah came to be recognized as being relevant/important because of the NT writers.