If the enterprising soul were to review a Biblical covenat such as the Abrahamic, that individual would discover that first Jehovah cuts the covenant (gen ch 15 vs 1..) then he confirms it (gen 15 vs 9) then shows the sign (gen 17) However complete fufillment still to come.
The OP question is:
did the new Covenant Start in Gospels or Acts?
A better question is when was/will the new covenant be fully instituted?
Here are the terms of the New Covenant:
Jer 31:31 "Behold, days are coming," declares the LORD, "when I will make a new covenant with the house of Israel and with the house of Judah,
Jer 31:32 not like the covenant which I made with their fathers in the day I took them by the hand to bring them out of the land of Egypt, My covenant which they broke, although I was a husband to them," declares the LORD.
Jer 31:33 "But this is the covenant which I will make with the house of Israel after those days," declares the LORD, "I will put My law within them and on their heart I will write it; and I will be their God, and they shall be My people.
Jer 31:34 "They will not teach again, each man his neighbor and each man his brother, saying, 'Know the LORD,' for they will all know Me, from the least of them to the greatest of them," declares the LORD, "for I will forgive their iniquity, and their sin I will remember no more."
Jer 31:35 Thus says the LORD, Who gives the sun for light by day And the fixed order of the moon and the stars for light by night, Who stirs up the sea so that its waves roar; The LORD of hosts is His name:
Jer 31:36 "If this fixed order departs From before Me," declares the LORD, "Then the offspring of Israel also will cease From being a nation before Me forever."
Jer 31:37 Thus says the LORD, "If the heavens above can be measured And the foundations of the earth searched out below, Then I will also cast off all the offspring of Israel For all that they have done," declares the LORD.
Jer 31:38 "Behold, days are coming," declares the LORD, "when the city will be rebuilt for the LORD from the Tower of Hananel to the Corner Gate.
Jer 31:39 "The measuring line will go out farther straight ahead to the hill Gareb; then it will turn to Goah.
Jer 31:40 "And the whole valley of the dead bodies and of the ashes, and all the fields as far as the brook Kidron, to the corner of the Horse Gate toward the east, shall be holy to the LORD; it will not be plucked up or overthrown anymore forever."