I was unfortunately out of town when this was first posted. I hope you still find it useful.

By summary, the Mormons largely sat out the war, and that seemed to suit both Brigham Young and Abraham Lincoln.

When the war first broke out, one of Lincoln's first priorities was to establish and maintain communication with the west coast. With the southern overland routes closed, Utah loomed large. The telegraph line to Sacramento was completed in October 1861. Interestingly Lincoln didn't bother asking the Utah governor to raise men to protect it. Instead he asked Young to provide 100 cavalrymen, which Young promptly did. These 100 men accounted for over half the 197 Mormon men who served in the Civil War.

Lincoln clearly didn't trust them to provide such functions long term, and less than a year later they were replaced by 700 volunteers from California, who camped in the hills right above Salt Lake and named their fort, "Douglass" after the great Mormon enemy Stephen Douglass, a very provocative act. By the end of the war, the fort had grown to several thousand men who largely participated in anti-native campaigns, including the Bear River Massacre. Around the same time. Lincoln turned down an offer of up to 10,000 Mormon volunteers, clearly showing a lack of faith that Young could deliver, or that such men should be trusted.

Lincoln's attitude can be best summed up by a statement he gave to T.B.H. Stenhouse in 1863:

When I was a boy on the farm in Illinois there was a great deal of timber on the farm which we had to clear away. Occasionally we would come to a log which had fallen down. It was too hard to split, too wet to burn, and too heavy to move, so we plowed around it. You go back and tell Brigham Young that if he will let me alone I will let him alone.

Lincoln didn't want to engage in questions of Utah, Mormons, or polygamy. He simply wanted to postpone any discussion on any of them. That seemed to suit the Mormons just fine.

Lincoln never met Young's request to appoint Young as governor of Utah. And his first two appointments went quite awry. They and the first judges were fairly antagonistic toward the Mormons, so neither lasted very long. But Lincoln learned from those and eventually appointed James Doty, who became popular with both the LDS and non-LDS people in the territory.

Similarly Young and the Mormon's attitudes toward the federal government remained tempered through most of the war. Young heartily disliked the Republican Party. He was well aware of their plank to cut out the relic of barbarism, polygamy. The arrival of Colonel Conner's Californian troops ratcheted up tensions. As did the passage of the Morrill Anti-Polygamy Act. The realization that the Act was largely unenforceable, and that Lincoln had no intention of informing it, calmed things back down.

The Mormon experience in Missouri, Illinois, and the Utah War of 1857-1858 left deep marks on the Mormon psyche. They were plenty happy to sit the Civil War out. Lincoln seemed quite satisfied to let them sit it out. When he died there was truly a sense of mourning in Salt Lake. Most businesses closed, and a service was held in the Tabernacle for 3,000 mourners. Lincoln's plan to "let Young alone," paid major dividends and worked to reintegrate the Mormons back into mainstream American society. This was then tested in the next 40 years of turmoil over polygamy, but that's a different question entirely.

This was largely drawn from Civil War Saints (ed. Kenneth Alford), particularly the Lincoln chapter. You may also find some of the following useful.

E. B. Long, The Saints and the Union: Utah Territory during the Civil War

Ralph McGinnis and Calvin Smith, Abraham Lincoln and the Western Territories

Leonard J. Arrington, Brigham Young: American Moses