According to Reformed theology, when did the New Covenant start? When instituting the Lord's Supper, Jesus Christ said (Luke 22:20, NIV):


  In the same way, after the supper he took the cup, saying, “This cup
  is the new covenant in my blood, which is poured out for you.


And in Jeremiah 31 the new covenant is also mentioned, with the blessings of the covenant listed.  Hebrews 8 talks about the new covenant superceding the old in some way (the details of which perhaps I will put in another question). 

At which stage did this new covenant supersede the older covenant, according to Reformed theology?
Reformed theology's view of the &quot;covenants&quot; of scripture is that they are administrations of a single covenant, a covenant of grace.  Thus the &quot;new covenant&quot; that Christ talks about is seen as something that in many ways had existed long prior.  As Louis Berkhof writes:

The covenant of grace, as it is revealed in the New Testament, is essentially the same as that which governed the relation of Old Testament believers to God.  [...] If it is sometimes spoken of as a new covenant, this is sufficiently explained by the fact that its administration differs in several particulars from that of the Old Testament. (Systematic Theology, 2.3.5.C)

The differences that Berkhof sees in the &quot;new covenant&quot; are its universal nature, its greater emphasis on grace, and its richer blessings.  But in essence it is the same covenant as that given to Abraham, Moses, and others.  So strictly speaking, while the &quot;new covenant&quot; begins in the New Testament with the life, death, and resurrection of Jesus, it is essentially the same covenant as the &quot;old covenant&quot; that preceded it.
Robert Reymond agrees, and quotes John Murray:

The blessings of the covenant of grace which believers enjoy today under the sanctions of the New Testament economy are founded upon the covenant which God made with Abraham.  Said another way, the &quot;new covenant&quot; itself is simply the administrative &quot;extension and unfolding of the Abrahamic covenant.&quot; (ST, ch 14)

Reymond refers to Hebrews 8 to argue that the Mosaic tabernacle foreshadows the &quot;true, heavenly Tabernacle into which Christ himself entered with his own blood as the redeemed man's High Priest.&quot;  Reymond specifically sees the incarnation and crucifixion as foundations of the &quot;new covenant&quot;:

Christ's &quot;entrance into the heavenly sanctuary&quot; occurred when he assumed his high priestly role as Mediator of the new covenant at the incarnation, and the Most Holy Place was his cross! (ST, ch 14)

John Frame helpfully explains how this &quot;new covenant&quot; is &quot;time transcending&quot; – how it has beginning in Christ but its efficacy extends into the past:

The new covenant does have a temporal inauguration.  Covenants are typically inaugurated by the shedding of blood, and that is certainly the case with the new covenant, by the blood of Christ, the blood that fulfills all the blood of bulls and goats in the other covenants. [...]
Nevertheless, as we saw earlier, the efficacy of the new covenant, unlike that of previous covenants, extends to God's elect before Jesus' atonement.  When believers in the OT experienced &quot;circumcision of the heart,&quot; or when they were Jews &quot;inwardly,&quot; they were partaking of the power of the new covenant. (ST, 80)

So to summarize – Reformed theologians see the &quot;new covenant&quot; as an administration of the more general &quot;covenant of grace,&quot; and in essence identical to the covenants laid out in the Old Testament.  But while it applies to all of redemptive history, the &quot;new covenant&quot; has its historical inauguration in the life and particularly the death of Christ on the cross.