There were Nestorians in Korea before the Mongol conquest, so I suppose a mass conversion is possible, though it is difficult. There were communities of Nestorians scattered throughout Central and East Asia, but no polity adopted it wholesale.
Part of that has to do with a lack of institutional support. They just weren't as organized as the religions that had local roots and so when it came time for groups to choose who would administer their state, the Nestorians tended to lose out after Wuzong.
In my TL, the Naimans became somewhat stronger and ended up becoming more fully Nestorian Christians. About a century later, Nestorians from other areas began experiencing some persecution from the rise of the militant Buddhists. Eventually most of these Nestorians placed themselves under the Naimans. However they could not stand alone against them and were eventually subordinated by a Jurchen (alternate Jinn) group. Placed on the border against the Buddhists, they were supported by their overlords who eventually took over Korea as a military aristocracy (this
almost happened in OTL) instead of China. They began to intermarry. Nestorians are quietly filtering into upper Korean society in the 1100s, it's seen as a kind of an edgy thing so far. Far from a majority but a strong minority and for the moment at least, in partnership. That there is no Neo-Confucianism to reform that philosophy also helps, and Far East Nestorianism has also had some interesting changes, but even in OTL it should be noted that the far east churches did not really talk about "2-natures" much.