The onset of the New Covenant meant the full implementation of God's undertaking to Noah (Gen 6:18) to maintain his covenanted purposes for the New Creation (Gen 1:26-28; Gen 2 - see Dumbrell 2002). I note that the use of heqim berit (Gen 6:18a) in the OT always points to maintenance, not commencement. Moreover, Gen 6:18b with 6:18a indicates that the salvation of Noah is that maintenance of purpose for creation. 'Covenant' from Gen 6:18a onwards becomes the language by which God's promise structure for history proceeds. That a divine intention to bring to a conclusion the work commenced with creation should be given the title of 'covenant' should not surprise. Biblical covenants are divine promises unilaterally imposed, firmly backed by covenant arrangement.
Jeremiah's New Covenant which Jesus' death and resurrection inaugurated, but not completely implemented until the Parousia, meant the dismissal of disobedient national Israel's election for service and the end of her institutions: law, temple, sacrificial atonement etc. Divine creational law (cf. the Decalogue), however, continues with a general obligation for all and to be written in the heart of believers (OT and NT). Jesus' New Covenant meant the onset of the New Creation age begun by the appointment of a New Israel (cf. John 20:22 correlated with John 1:12).
The New Covenant was thus the implementation of the Abrahamic Covenant of which the Sinai Covenant with Israel had been a subset. Paul's New Covenant ministry (2 Cor 3:6) that recognized all of this (cf. Rom 6:14, Gal 2:23, 2 Cor 3:6) confronted Jewish Christians (cf. 3:6 - note the present of 'kills') who saw Jesus as operating within the continuing Mosaic Covenant.
This was the problem facing Paul in Galatians, Romans and 2 Corinthians and we may see traces of it elsewhere in the Pauline Epistles. Paul's in his appraisal of Israel carefully recognized the changed position resulting from the cross (cf. Rom 2:1-3:20; 9:30-10:8; 7:1-6, 6:14, etc.).
The general reluctance of NT scholarship to accept a covenant emphasis, in view of Jesus' action and the Jewish character of the early church is puzzling. It is an undervaluing of how OT theology of kingdom and covenant works its way through the whole Bible. The usual objections of lack of reference to the terms, apply to the OT as well as to the new but the notion is basic to the correlation of the two Testaments/Covenants.