While I don’t think we can pinpoint the exact moment when God declared Jesus our High Priest, I believe the Book of Hebrews gives us an idea of a general time period.
Hebrews 7:20-21 tells us that Jesus was made our high priest by an oath sworn by God the Father.
Hebrews 7:20-21 (KJV):
20 And inasmuch as not without an oath he (Jesus) was made priest:
21(For those priests were made without an oath; but this with an oath by him that said unto him, The Lord sware and will not repent, Thou art a priest for ever after the order of Melchisedec:)
So the question is when did the Father make this oath?
We understand that the Old Testament Covenant of the law required a human priesthood. That priesthood was created by God at the giving of the Law of Moses. That priesthood was established in Aaron (the brother of Moses) a direct descendant of Levi, son of Jacob. All priests that served God thereafter, were to be of the Tribe of Levi as required by the Law of Moses.
However, the Book of Hebrews goes on to tell us that when the Father began to lay the foundation for the New Testament Covenant of Grace, God instituted a change in the law and in the priesthood that would be able to deliver all mankind from their sin. The priesthood was then changed from the Tribe of Levi to the Tribe of Judah, the tribe where Jesus descended.
Hebrews 7:11-14 (KJV)
11 If therefore perfection were by the Levitical priesthood, (for under it the people received the law,) what further need was there that another priest should rise after the order of Melchisedec, and not be called after the order of Aaron?
12 For the priesthood being changed, there is made of necessity a change also of the law.
13 For he of whom these things are spoken pertaineth to another tribe, of which no man gave attendance at the altar.
14 For it is evident that our Lord sprang out of Juda; of which tribe Moses spake nothing concerning priesthood.
Hebrews 7-28 then tells us that the oath that the Father made that gave the title of High Priest to Jesus, was made after the Law of Moses was given to Israel.
Hebrews 7:28 (KJV)
For the law maketh men high priests which have infirmity; but the word of the oath, which was since the law, maketh the Son, who is consecrated for evermore.
This makes logical sense since Hebrews 7:12 declares that there was a “change” to the original Levitical priesthood plan, logic states that it would be “after” the Levitical priesthood was established. Also, (while I am no Greek student) Thayer’s Lexicon states that the usage of the Greek “meta” (ie, the word “since” in Hebrews 7:28) has the sense of time sequence, ie “after”, ie after the Law of Moses was given.
Finally, Psalm 110 references this same oath.
Psalm 110:1-4 (KJV)
The Lord said unto my Lord, Sit thou at my right hand, until I make thine enemies thy footstool.
2 The Lord shall send the rod of thy strength out of Zion: rule thou in the midst of thine enemies.
3 Thy people shall be willing in the day of thy power, in the beauties of holiness from the womb of the morning: thou hast the dew of thy youth.
4 The Lord hath sworn, and will not repent, Thou art a priest for ever after the order of Melchizedek.
So, I believe we can place the establishment of Jesus as High Priest somewhere between the giving of the Law of Moses (circa 1500BC) and the time Psalm 110 was written, (circa 1000BC).
Some Protestant Christian Commentaries agree with the timing of the oath (ie "after" the law was given) but also believe the "execution" of the oath (seeing it as a promise) was not realized until after the death and resurrection of Christ. Here's Matthew Poole's commentary on Hebrews 7:28:
"But the word of the oath, which was since the law; but God the Father’s promise to his Son, ratified with an oath, that he should he the great High Priest perfecting of souls for God, as David testifieth, Psalm 110:4, to be revealed to him; and this four hundred years after the law was given which constituted the Aaronical priesthood. The word revealed God’s promise to him, the oath made it irreversible; yet this promise was not actually performed to him till his ascension in the human nature higher than the heavens, Psalm 110:1."