This possibility was explored in the Deadlands series of tabletop roleplaying games. In that series, Desert was restricted to the modern-day state of Utah.
Let's remember that any discussion about Deseret continuing to exist means that we have to assume that other changes had to have happened in the timeline. These are the changes to history that Deadlands assumes that affect Deseret.
- The US Civil War petered out into a stalemate, with the result of the Confederate States of America remaining independent
- Increased interest in a trans-continental railroad resulted in the CSA annexing a good portion of the southern Great Basin
- The Union Army had several tragic armed conflicts with the Mormons (it was never all-out war, and I vaguely recall that the US Army was the aggressor in those cases)
- Neither had the resources to annex Deseret and it was deemed a low priority.
Also, the Mormons felt that neither government was serving their interests and they were better off remaining independent than voluntarily joining either nation:
The Saints muddled along until 1866. By then, with no end in sight for the Civil War [thus no support from either government who were too focused on the war] and several unfortunate conflicts with Gentiles (including the Union army) under their belts, Brigham Young declared the Mormons would rule themselves until such time as the government had the resources to properly address the Saints’ concerns and complaints. The new nation was named Deseret, and encompasses all of the territory known to Gentiles as Utah. —Deadlands Reloaded: Player's Guide
In other words, while the Mormons might have wanted to rule a larger area, conflicts with the United States government and annexations from the Confederate States government resulted in the Mormons choosing to have Desert be a smaller than planned area in order to avoid further conflicts with either government. Neither the USA nor the CSA had the resources to pursue the matter further at the time, so they became a de facto independent nation.
In Deadlands Noir, set in the 1930s, this is what the map looked like (with the Confederate States of America still extant, along with the Sioux Nations):
The nation is also still present with the same boundaries prior to the apocalypse that begins Deadlands: Hell on Earth, set in the 2090s. So Deseret maintained sovereignty for over 200 years.
To my knowledge, this is one of the most thorough explorations in a work of fiction concerning how an independent nation of Deseret might have developed. Things might have developed differently had there been different changes to the original timeline than those outlined above, but I think that this is a reasonable interpretation given the timeline of events that the authors presented.