The Temple Institute has already recreated all the implements for the temple, including the clothes of the priests, and has trained the priests for the daily Temple ceremonies.
As well, they have at least created the "corner stone" (if not the entire temple), and architectural blueprints and computer models. As well, the Sanhedrin has reformed.
The Performance of the Torah's Commandments Does Not Depend Upon the Messiah
There are no Biblical verses that make a connection between the building of the Holy Temple and the arrival of the messiah. The building of the Holy Temple is a commandment that is binding upon all of Israel to fulfill, in every generation.
But it must be noted that there is a difference between the commandments that Israel are obliged to perform, and the prophecies referring to the appearance of the messiah. The prophecies are a Divine promise; a heavenly revelation concerning the appearance of the redeemer in the end of days.
Nothing can ever change that Divine promise. However it is important to understand that the arrival of the messiah is not dependent on anything other than the will of G-d alone. He shall fulfill His promise at the time that He deems fit.
In fact, if there really is a question as to "Which comes first, the messiah or the Temple," there seems to be ample indication that the building of the Holy Temple will precede the messiah's arrival.