A Nestorian Mongolian empire would have been quite interesting indeed, particularly regarding to the Crusades.
For instance, when the Mongols first showed up in the Near East in the mid-1200's, the Pope actually sent a delegation -which included a hefty number of Dominican and Franciscan friars- all across Asia to meet the Great Khan at his court in Qaraqorum, in an attempt to covert him to Catholicism. However, he responded by demanding the Papacy to submit to Mongol domination.
Christians initially hoped that the Mongols might prove allies in defending the Holy Land (which by then was on the brink of collapse on the hands of the Mamluks), mostly because people rumoured they had all become Nestorians (even though that particular branch had been declared heretic after the Second Council of Ephesus, Christian is Christian). Some even believed they were the armies of Prester John, rallying from his mythical realm on the east to save Christendom in its darkest hour.
Had the Mongols actually been Nestorians, or had the papal legates managed to convert the Great Khan to Catholicism, the history of the Near East would have been tremendously different, as they would have most likely allied with the Christians of Outremer and together repel the Mamluk attacks.
At the same time, one has to wonder how would have the relationship between Mongols and Europeans have shaped should the former have been Christians. After all, even after the Golden Horde had officially converted to Islam, clashes between Mongols and Muslims raged on for a long time. I'd imagine Christian Mongols would have perhaps been a bit more forgiving with Poles and Germans, but I'm not sure they would have stopped conquering everything in their path. Perhaps such conversions would have even made them more keen on conquering Europe, under a pan-Christian banner.
Still, given the tendency of Mongols to fracture down into smaller, mostly independent realms, there would still be the question of whether or not the likes of the Persian Ilkhanate (the most powerful Mongol state in the Middle East by the 1300's) would have acted any different from what they did. After all, they mainly fought Muslims, not Christians, so even as Nestorians I think they would have acted pretty much the same.
A more far-flung idea would be that, as Nestorians, the Mongols would have sought some kind of legitimacy over the rest of Christendom, perhaps seeking to control the likes of Constantinople or Rome in search for domination as the true Christian faith, in contrast with how history turned out (as in they merely sending threats of subyugation to the Pope and other western rulers, but never really doing much beyond that).
In any case, a very intersting subject to work with!