The first issue I see about a Sassanid Nestorian Persia is the association that is made between Christianity and Rome.
While Persians can support Nestorians as a rival group of Christians, hoping to weaken the orthodox imperial position, I doubt they'll associate themselves with this and would likely persecute them as OTL.
However, it seems that Nestorians still were quite powerful before Islamic conquest and nothing really prevent its rise if associated with a more important iranisation (and maybe zoroastrisation) of its institutions and beliefs.
However, for it being widely adopted, you'll need a pretender or a sucessor to imperial throne being himself Nestorian and such thing would likely push a civil war that could durably separate west and east of Persia.
I can't see Turks carrying Nestorianism and imposing it on Persia however : as long they were steppe people, they could be tengric, Christians, Muslims or whatever they pleased (While the huge use of Turkic mercenaries by Islamic rulers would anyway be a push for Islamisation), but once they ruled a population, critically in the Caliph's name...I don't see how it could be conciliated.
Agreed. Persians, being rivals to the Romans/Byzantines, will want to be able to push a thorn into the Roman's side by supporting a disenfranchised old orthodox sect. This would greatly strengthen the Nestorian position and prestige, at the expense of course of relations between Persia and Rome.
Note that (according to Wikipedia):
"The Church of the East developed from the early
Assyrian Christian communities in the
Assuristan province of the
Parthian Empire, and at its height had spread from its
Mesopotamian heartland to as far as
China and
India." (
Assyrian Church of the East)
However, on the page for Nestorianism, we find:
"
Following the exodus to Persia, scholars expanded on the teachings of Nestorius and his mentors, particularly after the relocation of the
School of Edessa to the Persian city of
Nisibis in 489 (where it became known as the
School of Nisibis). Nestorianism never again became prominent in the Roman Empire or later
Europe, though the diffusion of the Church of the East in and after the 7th century spread it widely across
Asia. But not all churches affiliated with the Church of the East appear to have followed Nestorian Christology; indeed, the modern
Assyrian Church of the East, which reveres Nestorius, does not follow all historically Nestorian doctrine.
Despite this initial Eastern expansion, the Nestorians' missionary success was soon deterred. David J Bosch observes
By the end of the fourteenth century, however, the Nestorian and other churches—which at one time had dotted the landscape of all of Central and even parts of East Asia—were all but wiped out. Isolated pockets of Christianity survived only in India. The religious victors on the vast Central Asian mission field of the Nestorians were Islam and Buddhism...
[2]"
(
Nestorianism)
Finally, it should be noted that "Nestorianism was officially anathematized, a ruling reiterated at the
Council of Chalcedon in 451. However, a number of churches, particularly those associated with the
School of Edessa in Mesopotamia, supported Nestorius—though not necessarily the doctrine ascribed to him—and broke with the churches of the Roman Empire. Many of Nestorius' supporters relocated to Sassanid Persia.
[3][9] These events are known as the
Nestorian Schism." (
Church_of_the_East#Nestorianism)
The question therefore, is whether the strength of the Nestorian church--increasingly divided at this point as we saw--can sustain this growth and become so prominent as to warrant a full conversion by Persia, though a rival to Chalcedonian (and later Nicean) Orthodox Rome.
ChristianMartyrsandCensus
From the graph on page 3, it can be observed that compared to the Oriental Orthodox Assyrians, Copts, Ethiopians, etc. the Nestorians (now known as the Assyrian Church of the East, though the Assyrians reject the Nestorian label) are relatively weak--in fact, it appears that their largest number was just before a mass martyrdom contemporary with the East-West schism of 1054. Therefore, the only way Nestorianism would gain momentum and succeed in Persia, like a previous poster has stated, would for it to be Zoroastrianized (though this is fairly easy, due to the already significant Near Eastern influence on the three Abrahamic religions) and for a ruler/member of the ruling class to convert to Nestorian Christianity.
However, as previously stated, this would likely trigger a civil war; without any allies (which, as a Nestorian state, is highly unlikely, unless one were to hire Armenian or Mongol mercenaries), the war would be unwinnable/become a stalemate (though highly favourable to the Zoroastrian candidate who has the backing of the majority of the populace) further weakening the Persian state. A possible compromise settlement would have to be achieved, possibly similar to the Peace of Augsburg, instructing the entire population to convert to the religion of its ruler, leading to massive population redistributions and a possible increase in martyrdom rates. Persia would be further weakened against either a Roman reconquest of Mesopotamia, an Arab conquest of Persia (if we are to indeed assume the rise of Islam takes place as in OTL), or the worst case scenario for the Persian state--both. In either case, it is highly unlikely that Nestorianism would survive as a large minority religion, let alone that of the majority or ruling class in Persia, even if the (highly unlikely) situation of a royal conversion does manage to take place.