Model of the Temple Mount of the Gentile, Conrad Schick from 150 years ago that may provide us with useful facts and the view of “Minchat Chinuch” (Rabbi Yosef Babad): It is known that the Torah will not rely on miracles for commandments and laws.
A translated excerpt from the article by Pinchas Abramovitch on special places in the Temple Mount and models of the Temple Mount and the Temple of Conrad Schick.
Abramovitch claims that Schick abandoned missionary activity in favor of the study of the Land of Israel and Jerusalem. He produced models of the Temple Mount 150 years ago. The first model of the Temple Mount was built by Schick in 1872 at the invitation of the Turks. The Turks asked Schick to build a model of the Temple Mount that would be exhibited in a world exhibition in Vienna in 1873.Schick built a large model of the Temple Mount (on a scale of 1:200) that includes the mountain walls and all the structures on it. The luxurious and sophisticated model was built as a modular model – layer over layer with each layer can be raised by buttons. The underground levels of the Temple Mount are revealed among the various strata. Any of the structure on the Temple Mount can be lifted and underneath it is revealed a previous layer.
The full original Hebrew article by Pinchas Abramovich is found at http://har-habait.org/articleBody/31500

Minchat Chinuch: “Do not rely on miracles to build the Temple or to renew the sacrificial service”.

 

Minchat Hinuch Mitzvah 95
Now behold the Rambam wrote here, the structure of the Menora, and the structure of the Shulchan and the structure of the Golden Altar, see there. Yet the structure of the Ark and the vessels he did not write and it is obvious that all of these vessels were each time different. In the 2nd Temple they were not of the vessels of the 1st Temple and in the days of the Hasmoneans all the existing vessels were lost, as it became clarified that they made at the outset a Menora of iron, became richer, {and made it from more expensive metal} .
If so, we have to know the laws. And even today it is possible that if the kingdoms grant permission to build the Temple, it is a mitzva to build it as was clarified in the midrash that in the days of Rabbi Y. Ben Chananya they gave permission and they (the Jews) started to build.
And see the book Capot Temarim that Rabbeinu Netanel {comment, my version of Caphtor Vapherech, says it was Rabbeinu Yechiel of Paris} one of the authors of Tosafot wanted to come and offer sacrifices in Jerusalem and if so, this is true also to build the Temple (even though in Rambam’s opinion this is not so, and it will be clarified elsewhere). However, (regarding) the Ark immediately since the Testimony was hidden away and will not reappear until the coming of the redeemer, let him speedily come in our day , they did not make another {replacement} Ark outside of what Moshe made in the desert and when the redeemer, let him speedily arrive in our day comes, then the Tablets and the Ark will be revealed; if so there is no need to write the laws of the Ark.
Nevertheless, this needs investigation, for it is known that the Holy Torah does not use miracles for the performance of commandments and laws; see the commentaries on the Torah and at the forefront the Ramban {Nachmanides}. If so, definitely it is a mitzva for the generations, as the Rambam counted, this mitzva to carry it {the Ark} on the shoulders and also the prohibition not to remove the poles, even those things won’t be until the arrival of the redeemer, for now there is no Ark in our great iniquity and he wrote concerning what was during the time of the Temple and will be in the time of Mashiach; if so the laws of the Ark were also applicable during the first Temple period, for example, if the Ark was broken (and although it was the work of G-d, nevertheless, The Torah does not make the laws dependent on miracles); and even in the future it is possible that if the Ark be broken, it is definitely a mitzva to make an Ark for the (Tablets of) Testimony (Luchot, in Hebrew}. And in truth this did occur in the days of Eli that the Ark was taken captive by the officers of the Plishtim (Philistines) and it was there 7 months, see Shmuel (Samuel). And it is explained in Agada that Shaul (Saul) snatched the Tablets from the hands of Golyat (Goliath) and if so they definitely needed to make then a new Ark for the sake of the Tablets.

Minchat Chinuch brings another precedent that tells us not to depend on miracles regarding the Temple. He then concludes, if so we have to know the structure of the Ark, like all the Temple vessels, what details, if lacking, invalidate the vessel and which details do not invalidate.

There are a number of possible scenarios that predict in which manner the Temple will be built in the future. Here I will present a scenario that is not so well known, without predicting if indeed it will actually occur.

Rabbeinu Bacheyei brings this scenario down in his commentary to Vayikra/Leviticus Chapter 11

And Midrash Tanchuma contends, why was this kingdom compared in a parable to the Chazir animal. For in the future the Holy One Blessed be He will put back on them (in Hebrew, LehaChazir) the strict attribute of justice. There are other versions in which it is written in them: That in the future it will put will put back (in Hebrew, LehaChazir) the crown to its old position. And the matter is that the 2 Temples were built by Israel, the First Temple, Shlomo (Solomon) built, who was from the seed of Yehuda (Judah), the 2nd Temple, Zerubavel built, who was also from the seed of Yehuda (Judah) for it was stated (Zechariah 4:8) “the hands of Zerubavel have founded”, etc. and it was done by Coresh (Cyrus) who gave permission in this matter; but the 3rd Temple, in the future will be built by this nation. And that is what they stated: “That in the future it will put will put back (in Hebrew, LehaChazir) the crown to its old position”. Because this one (kingdom) destroyed it. {end of quote}

In Jewish thought, Christianity generally came to be associated with the Kingdom of Rome, because that religion became the official religion of the oppressive empire. I do not know if this Midrash believed the biological descendants of Rome would restore the Temple or did it mean the religion adopted by Rome would restore the Temple to the Jews.