This could have some interesting ramifications. Its been a while since I've read up on the Church of the East, but the Patriarch in OTL tried to form a firm alliance with the Mongols. When the Khans went Muslim instead, it actually caused the Church of the East to go into steep decline (previously they had been largely tolerated by Muslim rulers, but the power play showed that they could be a threat). The Nestorian sympathies of the Mongols also excited the Pope in Rome who tried to establish a diplomatic relationship; although they might be heretics, at least they were Christians.
Now, I suppose just how successful the Church of the East would be in the ATL depends largely on the manner in which the Khans promote Christianity. If it becomes politically beneficial to be Christian in order to get promoted/gain prominence, I suspect that there will be waves of conversions. It will also be determined by just how long the Mongols manage to rule; Nestorian or not, if their rule is but a flash in the pan, then it won't have any noticeable impact.
Although the impact on the Middle East would be great, I would actually be really interested in exploring the impact on Russia too. Here you would have a subject Christian population, but one which is Orthodox. I could see the chance for a whole bunch of fun; especially if the Russian Patriarch ends up entering into communion with the Church of the East (although I have no idea how likely or unlikely that would be).