Definitely - the Church of the East was making huge inroads into Central Asia, and I would expect that also to be the case ITTL. Christianity, therefore, would have Persian-Nestorian elements here, with Baghdad as its major center, as well as part of a wider Asian world even stretching as far east as China (extra bonus points if the Church of the East remains the representation of Christianity in China and, like Buddhism, survived the ban on foreign religions) and as far south as India. As far as Oghuz Turks go, Christianity would continue to have a native Turkic substrate, and would only really have a go towards Anatolia if the Persians used Turkic peoples as soldiers, much like OTL after the conversion to Islam. Thus, what we would have - if they did migrate towards Anatolia - is an encounter of two very different Christianities, a Turco-Persian Nestorianism on one hand and a Greek Orthodox Chalcedonianism. Ideally, the ERE would like it if the Turks migrated away from what Constantinople would definitely as heretical and heterodox ideas and beliefs and conform towards the beliefs and teachings of the Greek Orthodox Church. In actuality, it would probably not be as clear-cut as that.