What it says on the tin, what would the affect be on Mormonism without the death of its founder?
Possibly no restriction of the priesthood to blacks. Brigham Young was adamant about racial restrictions, Joseph Smith was more pragmatic. In early Mormonism while Smith was alive, black men were ordained to the priesthood, though he distanced himself from racial inclusionary practices when the church was trying to survive in Missouri, a slave state. After leaving Missouri, he probably would have put the church on a trajectory towards fuller racial inclusion.
20 Behold, Potiphar’s Hill was in the land of Ur, of Chaldea. And the Lord broke down the altar of Elkenah, and of the gods of the land, and utterly destroyed them, and smote the priest that he died; and there was great mourning in Chaldea, and also in the court of Pharaoh; which Pharaoh signifies king by royal blood.
21 Now this king of Egypt was a descendant from the loins of Ham, and was a partaker of the blood of the Canaanites by birth.
22 From this descent sprang all the Egyptians, and thus the blood of the Canaanites was preserved in the land.
23 The land of Egypt being first discovered by a woman, who was the daughter of Ham, and the daughter of Egyptus, which in the Chaldean signifies Egypt, which signifies that which is forbidden;
24 When this woman discovered the land it was under water, who afterward settled her sons in it; and thus, from Ham, sprang that race which preserved the curse in the land.
25 Now the first government of Egypt was established by Pharaoh, the eldest son of Egyptus, the daughter of Ham, and it was after the manner of the government of Ham, which was patriarchal.
26 Pharaoh, being a righteous man, established his kingdom and judged his people wisely and justly all his days, seeking earnestly to imitate that order established by the fathers in the first generations, in the days of the first patriarchal reign, even in the reign of Adam, and also of Noah, his father, who blessed him with the blessings of the earth, and with the blessings of wisdom, but cursed him as pertaining to the Priesthood.
The structure of the church also might develop differently. When Joseph Smith and his brother Hyrum Smith were killed at the same time, there was a succession crisis because rules hadn't been formally decided on how the next prophet would be selected. Hyrum was Joseph's chosen successor, as the Assistant President of the Church. In the power vacuum, the Quorum of the Twelve Apostles became the acting leadership of the church and since then the eldest member of the Quorum becomes the prophet on the death of the last. If Joseph lives longer, and names a successor, and that successor completes his succession peacefully, then subsequent prophets will be given the formal power to choose their successor regardless of seniority rights in the Quorum. This would become important in the Twentieth Century, as the prophet may name a younger Apostle as his successor, giving him more time to lead the church, rather than passing the mantle on to another elderly apostle who will only have his twilight years to effect policy. It could make the Mormon Church more dynamic.
Assuming a long life for Joseph, he would probably push the church towards a more socially radical, communitarian trajectory. In his last few years especially, in Nauvoo, when he thought he and the church were safer than they'd ever been, he began revealing his most extraordinary commandments and revelations, such as the King Follett Discourse in which he claimed men could become gods, or the revelations concerning polygamy. Given decades of more time, you'll probably see much bolder, visionary leadership than what even Brigham Young was capable of.
Here's the thing, I don't think that the Church of Jesus Christ of Latter Day Saints would have survived had Joseph Smith Jr. survived. As time went on, Joseph Smith Jr. increasingly begun to develop his theology away from the vanilla protestantism he had originally presented in the Book of Mormon. Exaltation, polygamy, the Masonic-inspired temple ceremonies, etc. I don't think that Joseph Smith would simple have stopped developing his doctrines by the mid-1840s had he survived. Remember that the Book of Joseph that he claimed had accompanied the Book of Abraham in the papyri he purchased back in Kirtland still was "untranslated". Likely, he would have produced such a work as well, going more deeply into the themes he had begun sketching on in the Book of Abraham and the King Follett Discourse.
I am particularly intrigued by the Adam-God doctrine, enthusiastically preached by Brigham Young during the Utah years. Brigham insisted on that this was no invention of his own, but only what he had been taught by Joseph Smith, and took it so seriously as to include it in the so-called lecture at the veil in the early temple ceremonies. I don't see why we should doubt Brigham on this point, as Joseph was known to first only spread new doctrine to a few select before revealing it to the membership at large (see for example exaltation, polygamy, etc.). I don't think it's unlikely that a Book of Joseph would have explored this.
Then there's Smith's fascination with Masonry. The temple ceremony is filled to the brim with Masonic influences, and early Mormons taught that this was because Masonry had its origins at Solomon's Temple (another belief very common at the time), which Mormon doctrine teaches contained the very same rituals modern Mormon temples do. It had over the years however become "corrupted", and Joseph Smith Jr. merely restored them.
And then there are Joseph Smith's consistent failures with his communitarian and utopian designs, such as the United Order, the Kirtland Safety and Anti-Banking Society, Zion's Camp, etc. These were the failures that led to the most defections and apostasies during his lifetime. Unlike Brigham Young, Joseph Smith, Jr. was a terrible administrator and executer. He lack the shrewdness, the pragmatism and the sense for business.
And then as always, there is polygamy, which during Smith's time as prophet was practiced with the utmost secrecy, because many leading members of the Church were actually fiercely against it, such as Sidney Rigdon. And do not forget that Emma Smith hated the practice, and several times throated to leave Joseph and go east with their children back to her parents.
And then, there's Joseph's increasingly megalomaniacal personality, from his being crowned King of Israel by the Council of Fifty in 1844, to his campaign for the United States Presidency that very same year, when apostles were sent on missions to campaign for him.
No, if all this is allowed to continue brewing, we're looking at impending implosion. Polygamy will eventually make Sidney Rigdon leave the Church, with him denouncing Joseph. Emma will eventually say that enough is enough, and desert Joseph, returning east, possibly taking their children with her. The increasingly syncretic theology will soon enough start alienating even the most die-hard supporters such as Brigham Young, John Taylor and Heber C. Kimball. It will not surprise me if everything culminates in Joseph Smith, Jr. declaring that he is Christ reincarnated, and the whole superstructure ultimately collapses.
I strongly doubt that the Church will even survive the remainder of the 19th century. If it exists today, it would be in a form not dissimilar to the form in which the Strangite Church still exists today. Here's their website, by the way.
Now, let's move to some secular concerns. Consider Nauvoo as a large metropolitan area. Just as Louisville sits on the "falls" of the Ohio River, Nauvoo is on the Des Moines Rapids in the Mississippi, a hindrance to river commerce until the Keokuk hydroelectric plant, with its 37-foot dam, controlled the river in 1913. If Nauvoo grows and there is no migration to Utah, what happens with the upcoming transcontinental railroad? The settlement of the Rocky Mountains would be very, very different.Assuming a long life for Joseph, he would probably push the church towards a more socially radical, communitarian trajectory. In his last few years especially, in Nauvoo, when he thought he and the church were safer than they'd ever been, he began revealing his most extraordinary commandments and revelations, such as the King Follett Discourse in which he claimed men could become gods, or the revelations concerning polygamy. Given decades of more time, you'll probably see much bolder, visionary leadership than what even Brigham Young was capable of.