Well, to begin with, you'd need more Mormon nationalism. Most Mormons were perfectly fine with the Latin alphabet; it was what they were used to, after all. Only an intense hatred of both the American government and it's culture would make them want to switch.
The Deseret alphabet was also extremely expensive; the Wiki article points out that a library composed of stuff written in the Deseret alphabet would cost over a million dollars (in the 1840s/50s, when that was a mind-boggling sum).
I have three (or so) suggestions that would have to be followed to make this work:
1. You have to cause the Utah War to escalate, but have the US government unable to bring too much of the Army to subdue the Mormons. The Mormons would be
fantastic at guerrilla warfare (Utah has the right terrain, and the Mormons have sufficient motivation), but even they wouldn't be able to stand against all or even a large portion the US Army. Nationalism seems to increase with successful guerrilla warfare, so would aid the desire to be different from the hated Americans. Look at the
Wiki page for the Utah War for ideas on making the Utah War more violent.
2. You have to give the Mormons a great deal of money. And here's the thing; gold won't cut it. The gold will run out, and then the Deseret alphabet will die in it's cradle. Historically, the Mormons managed to prosper (somewhat; they were still pretty poor) from charging a pretty penny for supplies for the gold rushers; I'm thinking you'd need to lengthen the gold rush somehow, or cause a second rush for silver or whatever.
3. Get rid of the LDS support of the
White Horse Prophecy. Have Brigham Young denounce it as not being written by Joseph Smith or something. That's one of our biggest ties to American nationalism; we don't see ourselves as a fundamentally different nation, we see
ourselves as saviors of the American nation. (which, speaking as an liberal American Mormon, I think this spiritual nationalism is a mite ridiculous). Make the White Horse Prophecy discredited (probably during the height of the escalated Utah War), and you'll lessen patriotism towards the United States.
4. It's not needed, but what might help is cooperation with the Native Americans. If they send their children to Deseret schools, and want to learn Deseret writing, Brigham Young might be able to justify the expense and difficulty of the alphabet by pointing to the "Lamanites" (Book of Mormon term) desire to learn it. Perhaps the Native Americans get into a spat with the American government, and Brigham openly sides with them? Maybe you could even have them help in the Utah War. Side Note; a few Native American chiefs converted to Mormonism; they might be a factor in this.
Hope this helps!