The Old Covenant is the law given by Moses. The New Covenant is yet to be made with Israel. It is promised in Jeremiah 31:31-34; the time of it and the terms of it are explicitly laid down in Ezekiel 36:24-38.
The Old Covenant was one of demand, and though “ordained to life,” was in result one of condemnation and death (see 2 Cor. 3).
The New Covenant to be made with Israel, following on her deep repentance at the end of the great tribulation, and synchronizing with the personal reign of Christ in the Millennium, is one of pure sovereign grace, consisting of new birth, the forgiveness of sins, and the gift of the Holy Spirit. What a day will that be for Israel!
The righteous foundation for this New Covenant is already laid in the death of Christ, and though not ratified with Israel as a whole, it has been antedated in God's dealings in blessing with His saints from earliest times. Apart from new birth there can be no link with God in blessing. The forgiveness of sins and the gift of the Holy Spirit were alike promised in Old Testament times.
As to Christians, though not formally under it, they have the blessings of it already. The cup at the Lord's Supper signifies “my blood of the New Testament [Covenant]” (Matt. 26:28). The Apostle Paul and his companions were made “able ministers of the New Testament [Covenant]” (2 Cor. 3:6).
The Christian has, indeed, larger and fuller blessings than those of the New Covenant, but as the greater includes the lesser, so do Christian blessings include the New Covenant blessings.
Meanwhile these blessings are found in connection with the Church, but when the Lord comes for His people, and Israel is set up under Christ, as Priest and King upon His throne, the New Covenant will be made with Israel in a public way.