In 1837 the guy who who start the Taiping Rebellion had a vision that he was the younger brother of Jesus Christ after reading the Bible and suffering from a fever. What if he was introduced to Mormonism at the same time?
I do think a more successful Taiping is going to have some real effects on Mormonism as a religion--they have the same kinds of theological enemies and will be attracting the same kinds of criticism, and Mormon theology has features that would make Mormonism hospitable to some kind of outreach to the Taiping. Plus the elite Taiping are going to be polygamous in the Chinese tradition and this will further incline Mormons to see them favorably. Plus elements of the Taiping themselves are pretty eager for western approval and might be attracted to Mormon contacts for that reason. None of this is certain, or even highly likely, but its not unlikely.
Of course Mormons believed a lot of stuff the Taiping didn't and vice versa, but as far as I know neither group ever made a point of rejecting any belief that turned out to be important to the other group.
From what I know about both groups (Mormons, quite a lot, Taiping, less than I used to) I'd say that the big sticking points are going to be (1) the relative statuses of Joseph Smith and the Mormon prophets vis-a-vis Hong and his ATL successors in religious authority and (2) the idea that the Manchus were literally devils (in Mormonism devils have no flesh). (3) If Hong is literally claiming that there are 4 and only 4 members of the Godhead, the 4th member being him, Mormons would also object strongly to that. But as long as it gets interpreted as somewhat metaphorical, or at least non-exclusive, Mormson could probably accept it. The first two could probably be got around: you’d just say that each was an authority in their own sphere, or only formally subordinate one group to the other, and you’d resolve the devil problem by claiming that the Manchus were only servants of the devil or, at most, all literally posessed by devils. The 3rd would require the non-exclusive version of Hong's claim to be God's Second Son, or else call the claim merely symbolism or something.
One crazy thought: make the Taiping succeed in keeping at least part of China. I suspect that Mormon beliefs and Taiping belief can be made compatible easier than Taiping belief and traditional Christian belief--so thread the historical needle to have some kind of rappochement and doctrinal sorting out between the Taiping and the Mormons. Then have the Taiping pull a Meiji, which I think they are extremely apt to do. Bingo, a huge number of Chinese will be Mormon, with more becoming Mormon over time. Plus, Mormonism is now the successful model for native resistance to colonialism, so you might get Mormon movements elsewhere in Asia, in Africa, and even among Indians in Latin America (we Mormons would LOVE this last part). By the last half of the 20th Century, Mormonism is now, say, 5 to 6% of the world's population, several hundred million strong and still growing. That compares pretty well to the OTL Catholic percentage (17%), Protestants ( 9 - 10%), or Islam, (23%), or at least it puts Mormonism in the same conversation.
“Mormonism” is mostly even more conservative socially than OTL but probably more socialistic and anti-imperialist than OTL. It will have an OLd Left flavor.
However, while this worldwide Mormonism is recognizably Mormon, its going to be different from OTL Mormonism in significant ways. Its not going to be run out of Salt Lake. It will be a worldwide religion but not a worldwide church. Successful Taiping are not going to take directions from to an American Westerner prophet and vice versa, probably not even formally. I think a modus vivendi that looks a lot like Eastern Orthodoxy could result, in which each culture or nation has its own Mormon church. The doctrinal basis for this would be (1) the fact that the Book of Mormon has a separate church organization and even a separate set of apostles for the New World, (2) the many references in Mormon scripture to the Father and the Son having separate dealings with the scattered peoples of the world, (3) Mormonism's doctrinal flexibility and pragmatism, (4) the fact that Mormons have always emphasized the need for modern prophets who can deliver God's message tailored to modern circumstances can develop nicely into a belief that each nation, culture, civilizational area, whatever, needs its own prophet who can tailor God's message for it, (5) the somewhat flexible hierarchical arrangments in the Bible, where there are multiple prophets from different areas, and in the Book of Mormon, where you have an outsider like Alma setting up a church organization that the ultimate insider (the Nephite King) goes along even though he apparently has significant religious authority himself, etc.
The American Mormon church probably retains some first among equals status, but conflicts of authority are likely. This much more widespread Mormonism is also a Mormonism that is going to contend more with schism, mutual excommunication, and all that sorry stuff.
what kinds of interactions would the Taipings have with contemporary Christian movements that also had unorthodox ideas about additional scripture, continuing revelation, and the quasi-divine potential of certain members of mankind? In the Great War of the 1920s, would the most decorated units be federalized elements of the Nauvoo Legion, burning with the desire to punish the heretics who broke the Concord with Salt Lake in 1912 and who even had the temerity to 'excommunicate' the Prophet?
LDS missionaries were heading out to England, some European countries and island nations at this time period... China I'm not so sure about. I know they eventually got to China within a decade or two (even though they didn't have too much success), but 1837? Not so sure... that's only 7 years after the organization of the church.
If Hong meets Mormon missionaries I assume their unorthodox version of Christianity will appeal to him. He could very well be baptized, given the priesthood, made into a local Chinese leader... at which point I think he'll want more power and make some new 'revelations' that would elevate his position in the Church. Church authorities would excommunicate him, but he'd take his followers and create a splinter group.
There were a lot of new converts to Mormonism that shortly thereafter went off to create their own fanatical apostate sects. James Strang would be a key example. There was also a missionary sent to Hawaii in the 1850s or 1860s that, having arrived, declared that God wanted him to establish a new Church specifically for the Hawaiians. He began to merge local Hawaiian mythology with Mormon doctrine and called his own apostles and priests. The real apostles of the LDS Church had to mount an expedition to reclaim the Hawaiian members and convince them of apostate conditions.