Alright, a draft.
1833
For whatever reason, Joseph Smith does not establish the doctrine that the Constitution and the United States of America were designed by God.
1836 - 1838
Joseph Smith and Sidney Rigdon due not pursue the creation of a 'Mormon bank', the Kirtland Safety Society.
The Banking Crisis of 1837 hits the United States. Kirtland isn't as bankrupt as in OTL, but the populace is still wounded economically. In 1838, with the break-away of a few hundreds of members, Joseph Smith leads more than 1,000 members to regroup in Far West, Missouri. Around this time, the Danites are formed.
1839 - 1840
The Missouri Mormon War
With a slightly higher population, there are slightly more conflicts, but the end result is the same. The leadership of the Church of Christ is imprisoned for a short time, while the membership regroups in Nauvoo, Illinois along the Mississippi River. Let's say someone gets the idea to drain the swampland from the very beginning, instead of letting disease fester among the refugees. This would increase the amount of members.
In the Republic of Texas, Sam Houston dies of yellow fever (which did hit the city of Houston in 1839). Lamar, ever-popular Texan nationalist takes power.
I'm going to throw in this butterfly: at the Battle of Yungay, General Manuel Bulnes is defeated by Confederate soldiers. The Chilean-Argentinian victory in this battle was a fluke of history, they faced enormous odds. This means that the Peru-Bolivian Confederation is not dissolved.
1841 - 1843
Sidney Rigdon dies due to poor health (much earlier than in OTL, but he did have poor health at this time). He won't be challenging Brigham Young for leadership of the Church. Polygamy is taught for the first time. Nauvoo becomes a bustling city that in 1842 possesses a population as high as the capital, Chicago. I've calculated a total increase of about 3,400 souls from OTL. Joseph Smith becomes a martyr earlier during the following events: he is captured by Illinois militia in 1843, to be imprisoned for destroying an anti-Mormon printing press, but before he can be brought to jail, a lodge they are staying at is surrounded by an anti-Mormon vigilante mob, who kills the Prophet and the Illinois militia. Good propaganda piece. Brigham Young takes control a year earlier.
President William Harrison doesn't die of a cold and institutes the American System due to his loyalty to the Whig Party. There's a Third Bank of the United States and a high protectionist tariff (45% !) that leaves Northerners ecstatic and the Southerners furious.
Kenneth Anderson becomes the President of Texas. There is a successful expedition to Sante Fe to secure the Texan claim on the area. The Republic of Texas benefits from the high Harrison Tariff by opening up its economy to European investors.
Manuel Bulnes is never elected president of Chile due to his defeat in the recent war. Reformist Manuel Montt instead takes the presidency.
1844
A lot of exciting events get underway. Sam Brannan leaves with 330 Mormons from New York City to Yerba Buena where he will help develop the region and assert Mormon influence over the land. Brigham Young leaves for the Great Basin region and arrives that summer, three years earlier than OTL.
Van Buren runs against Henry Clay in the US presidential election. Clay wins re-election by a few electoral votes.
In the Republic of Texas, Lamarite John Richard Archer defeats Edward Burleson for the presidency.
1845 - 1849
Porter Rockwell, Joseph Smith's former bodyguard, leads an expedition throughout the Southwest to look for good colony sites. The Mormons will need such mapping projects. The Exodus begins in earnest. There are 42,000 members, compared to 30,000 of OTL.
The Native Americans in the Great Basin region establish good relations with the Mormons. Without the non-Mormon settlers of OTL, they will keep such a strong relationship. The Ute chief Wakara is baptized into the Church.
Mormon emigration into California invigorates the region. John Sutter founds the city of Sutterville on the site of his New Helvetia domain. OTL Las Vegas and the Inland Empire areas are the site of new budding Mormon colonies.
President Clay allocates $100,000 to settle free blacks in Liberia. Iowa becomes a state in 1846. Wisconsin becomes a state in 1848, upsetting the balance between free and slave states. Clay wins re-election against Lewis Cass when the Democratic Party is split by the Free Soil Party. His platform includes legitimizing the "unquestionable" American claim to the Oregon country.
Peter Bell defeats J. Pinckney Henderson in the Texan presidential election.
In the Peruvian-Bolivian Confederacy, Supreme Protector Santa Cruz forces the legislature to allow him an extra two years of rule, to 1849. Meanwhile, British investment into the guano mining industry begins yielding profits for the region. By 1849, the Supreme Protector Laws are issued to the Confederacy, outlining a system to select a Supreme Protector for a term of seven years. It gives the former Supreme Protector a great deal of power in deciding his heir. As such, popular liberal Ramon Castilla takes up the former mantle of Santa Cruz.
In Mexico, President Joaquin de Herrera is overthrown in a military coup. Ultraconservative ultranationalist Mariano Paredes y Arrillaga takes power. The Moderado faction is created from dissatisfied extremes of the Liberales and Conservadores. In 1848, Benito Juarez drafts the Plan de Ayutla, which causes a revolt against the Conservadores throughout Mexico. A year later, the Moderados take power and hold a Constituent Assembly, electing Ignacio Comonfort as President of Mexico.
The Yucatan declares independence once more in 1849.
1850
Yerba Buena has a population of 2,000, the majority of which are Mormons. It becomes a port used frequently by British merchants. In response to the Anglo population boom in Alta California, Mexico sends soldiers to assail Mormon colonies in the San Bernardino area. This agitates Mormons in Yerba Buena, who link up with local Mexican Liberales and declare intentions for a new constitution, though extremists voice desires for independence. Lands held by the Catholic Church is seized throughout the region. The Republic of Alta California is declared in Sonoma.
President Henry Clay dies, with VP Millard Fillmore taking his place. Border disputes over Oregon escalate: Fillmore announces a one-year notice to the British for the termination of the Oregon treaty. Neither side will compromise.
Peter Bell is re-elected as President of Texas, after a law is passed allowing Presidents of Texas to run for a second term.
The Mexican Civil War
Ignacio Comonfort and Benito Juarez declare the Constitution of 1850, including numerous liberal reforms and establishing religious freedom. Without the Mexican-American War, the Conservadores react even worse than OTL. However, five months later, President Comonfort throws Benito Juarez into prison, fearing a Conservadore coup. Even this does not sooth his worries, as in November he flees the capital after freeing all of the Liberale prisoners in Mexico City. Conservadore Felix Zuloaga, after being appointed president by an illegal Congress, seizes Mexico City and repeals the reforms of the new constitution. Benito Juarez creates a government in exile in San Louis Potosi.
1851
In the Columbian Compromise, slavery is abolished in Washington D.C., but a stronger Slave Fugitive Law is passed, placating the southerners just barely enough.
The date for coming to a resolution on the Oregon boundary dispute passes. The United States and the British Empire freeze the date to put the issue before international arbitration. Tensions are very high.
A race war between Mayans and European Yucatecos begins in the Yucatan.
The Deseretian Revolution
With news of a civil war exploding in Mexico City, movements begin throughout Alta California to secede from Mexico. In August, President of the LDS Church Brigham Young declares the Commonwealth of Deseret. That September, General Andres Rico begins the first battle of the Deseretian Revolution by leading a 200-man army to take Los Angeles, which succeeds.
In October, Salt Lake City emissaries arrive in Austin, Texas to plead for military aid. At the same time, they agree to partitioning certain Mexican territories.
In November, John Sutter discovers his workers have discovered large gold deposits in the nearby American River, and decides to keep it a secret until after independence is attained.
1852
There is still no conclusion to the Oregon Border Dispute. The US presidential election is seen as a referendum on whether to declare war against the British Empire once again. With the emergence of the American Liberty Party, Millard Fillmore fails to take the presidency, instead the Democrats finally seize victory under Lewis Cass. One of his first orders of business are to establish the Columbian Codominium, consisting of the area in contention, between the Columbian River and the 49th parallel, and Victoria Island, to be placed under the authority of a Codominium Commitee consisting of an equal number of American and British delegates. The joint rule would continue for 25 years until 1877.
The Mayan Cruzob cult begins a holy war against a communal native republic of the Yucatan just months after non-Mayans flee en masse from the Peninsula.
In February, a meeting in Sutterville between John Sutter and Brigham Young outlines a constitutional federal secular republic under the name of ‘the Commonwealth of Deseret’. The new territories claimed to be under this new country’s authority are split into two states: California and Zion.
In April, John Sutter publicly announces the discovery of gold in the American River after being unable to conceal it much longer. In May, pro-Mexican Californios and the Legion of Zion clash near the Colorado River, leading to a Mormon defeat. After this disaster, Brigham Young agrees to combine his two military forces: the Legion of Zion and the Legion of Deseret, with John Sutter’s National Army, creating the National Legion.
That summer, the National Legion pushes pro-Mexican forces down the Colorado river in a series of violent clashes. When a large Mexican army of 250 is spotted closing in on the San Diego area, the National Legion marches to cut them off. After securing a firm victory, the Treaty of New River is signed, ending the Deseretian Revolution with the Mexican confirmation of Deseretian independence.
In October, Mormon miners near Dalestown discover gold, prompting a gold rush from the Utah Valley. [This is the OTL Comstock Lode.] Meanwhile, the announcement of gold in California first reaches newspapers in New York City. By the end of the year, nearly four thousand gold-seekers arrive in Yerba Buena.
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I did a few calculations on the growth of the LDS Church with such different circumstances. Because there was virtually no succession crisis helped a great deal in boosting the numbers.
By 1850, ITTL there will be 81,000 members instead of 52,000 in OTL. By 1852, ITTL there will be 85,000 members instead of 53,000 in OTL. By 1860, I predict 116,000 members in the Mormon church instead of OTL 61,000... almost double. Why? Well, the Utah War caused a severe disruption in the Mormon Church, with the whole valley preparing to pick up and leave for Mexico. It scared members badly, preventing growth and causing 9,000 members to leave the Church in just one year. Replace that with a growth percentage rate average for the times... it is a significant increase in addition to all the others.