At the time of the publication of the first Sherlock Holmes novel in 1887 Polygamy was still publicly practiced in the Mormon church. It wasn't until the 1890 Manifesto that Polygamy "officially" ended under Wilford Woodruff. While it did not stop until years later it was better hidden and less open until the practice stopped altogether. Mormons were recruiting and proselytizing in the British Isles heavily since the religion had started under Joseph Smith. A lot of people were converted at the time. The Mormon movement under Brigham Young was very centralized, once you were converted you were expected to "Come to Zion" in Salt Lake City, later church officials would tell you where to go to settle the inter-mountain west. As such, many people were emigrating in large waves to the unsettled US from Britain following a strange rumored (but totally) Polygamist society. Was Doyle wrong about the polygamy? No.
Others I have ideas and speculation, most to do around the Mountain Meadows Massacre, but they are just that, speculation.
EDIT: Heh, Wilford Woodruff not Woodrow Willson
Quick note: I'll call Mormons "Latter Day Saints" (LDS) because the Mormon Church has had a complicated relationship with Mormon as a label for group identity. Second, what I'll say below is true for the followers of Brigham Young in Utah (known as the Rocky Mountain Saints, as opposed to the Prairie Saints who stayed behind in the Midwest after Joseph Smith's death) but they were not the only group within the larger Latter Day Saint Movement. The second largest group, which is only about 1/10 1/30th the size of the Salt Lake City-based Church of Jesus Christ of Latter-day Saints, is the Independence, MO-based Community of Christ, formerly known as the Reorganized Church of Jesus Christ of Latter Day Saints (RLDS). No other group ever achieved the popularity or notoriety of Brigham Young's faction so generally when people discuss Mormon's post-Joseph Smith, they're only discussing the Salt Lake City-based groups. I just want to acknowledge that there are other Mormon groups, and their relationship with plural marriage was not the same as the Church of Jesus Christ of Later-day Saints. For example, the RLDS church was opposed to the practice from its inception (and in fact its leaders have claimed in the past, with no convincing evidence, that Smith himself never practiced it).
So, to get to your actual question, there are accurate things about the story and inaccurate things about it. Many LDS churches at one point practiced polygamy. A few still do, but they're all tiny. These are generally called Fundamentalist Mormons (this term can either refer to the movement as a whole, or the largest denomination within the movement, the FLDS Church). Until 1890, the main Salt Lake City-based church still practiced "plural marriage", as polygamy is called in the LDS movement.
Joseph Smith, the founder of the first LDS Church, privately taught plural marriage during his lifetime. It's unclear when Smith began teaching plural marriage to his inner circle, but it's probably some time in the early 1830's. By the 1840's, the doctrine was more strongly established in the Church, but it was still not publicly acknowledged (in fact, Smith preached against it publicly). By the time of his death Smith had about three dozen plural wives, but the exact number isn't documented. Smith likely did not have sexual relations with all these women. Plural marriage, or "spiritual wifery", was the cause of the scandal that ultimately lead to Joseph Smith's lynching in 1844.
After Smith's murder, the movement split. The largest faction followed Brigham Young westward (the movement had been already moved several times, from New York to Ohio to Missouri to Illinois). The Utah Territory was organized in 1850 and Brigham Young was its first governor. In 1852, for the first time, LDS leadership publicly acknowledged plural marriage.
Parts of the Utah Territory were later broken off into all of Nevada and much of Colorado--mainly the areas that attracted large numbers of non-LDS settlers (mainly because of the silver rush in Nevada and a gold rush in Colorado). Utah itself was run as a quasi-theocracy (Young called it "republican theocracy") where Brigham Young was really in charge of both religious and political life. That part is also true, but I believe was changing by the time Conan Doyle was actually writing. Young died in 1877 and there was an organized, but if only occasionally elected, opposition in the Liberal Party starting in 1870. Secondly, while the Church controlled much of the politics in the territory, key positions, especially after the 1850's, like judges and the governor were in the hands of federally appointed officers. These federally appointed positions were generally held by people who were non-LDS and at times openly anti-LDS, as in the case of Eli H. Murray, who was governor of territory from 1880-1886.
I'm not sure exactly when the Utah parts of the story were supposed to be set. The story was written in 1886 and published in 1887. It depicts tense and at times violent relationship between Mormons and non-Mormons. The thing is, by the 1880's, these didn't so much exist anymore. The two many incidents that people point to--the 1857 Mountain Meadows massacre and the 1857-8 Utah War--were both a full generation earlier. There was really a paramilitary group called the Danites, as well, though I don't think they resemble the group in Conan Doyle's book very closely. By the time Conan Doyle was writing, this sort of paramilitary tension doesn't exist in the way it had in the 1850's.
I don't know about how apostates were handled, nor how relations with non-LDS members worked in the late Utah Territory. They were not like Conan Doyle's book, as much of what the country knew about the LDS church came from apostates like Ann Eliza Young. Ann Eliza Yoing went through a very public divorce with Brigham Young, in which he was initially ordered to pay alimony, but I don't remember her receiving threats or coercion. By the 1870's, there were several sensational stories of ex-LDS members being published back East. Some also went on popular speaking tours. I am unaware of cases where people were prevented from leaving or otherwise coerced violently by the church in this period. In fact, there were in fact several groups of people formally excommunicated by the Church who still operated in the territory, such as the Godbeites.
Continued below
I answered a related question in /r/latterdaysaints some time ago:
I am LDS and a reader of Doyle's Sherlock Holmes stories since childhood.
A Study in Scarlet is part of a long line of anti-Mormon literature and other media published in the US and Britain in the late 19th and early 20th centuries. These articles and books ranged from relatively factual accounts by disaffected members to bizarre slanders that would today be laughed at by anyone with common sense, but which many took seriously at the time. (As one example of the latter, in the 1922 British silent film Trapped by the Mormons the heroine escapes the evil Mormons' Salt Lake Temple by jumping out the window and landing in the Great Salt Lake. Anyone who has been to Salt Lake City will be able to tell you why this is a laughable notion.)
Much of the literature, especially in Britain, claimed that Mormon missionaries were actually white slavers who lured young single women to Utah for Mormon harems. Although such claims largely vanished in the US by the early 20th century--Utah was now a state, legally represented in Congress, and apparently not comprised of perverts--such claims continued in Britain for decades; Winston Churchill, for example, had to deal with demands that Mormon missionaries be banned from Britain during his time as Home Secretary. As Michael Homer explains, Doyle wrote about Mormons because 1) he had studied it and 2) writing about it, especially in a sensational manner, was good business at the time for a struggling doctor who wrote on the side for extra money.
Volume 2 of Leslie Klinger's The New Annotated Sherlock Holmes (2005), which I highly recommend you consult, offers an excellent and accurate explanation of relevant Mormon history in Scarlet's "The Country of the Saints". Klinger refers to Jack Tracy's Conan Doyle and the Latter-day Saints (1979) as the "definitive" examination of Doyle's treatment of Mormons. While I do not have access to Tracy's work, Klinger quotes from it. For example, "[The] suggestion that the Ferriers were forced to convert in order to be rescued is a patent misrepresentation of the Mormons' often demonstrated generosity toward their fellow travelers ..." Klinger later writes "Tracy asserts that no evidence exists that the [Danites] ever engaged in any activities in Utah" and, referring to the mention of a supposed funeral rite, "Again, there is no evidence of such a Mormon custom". In other words, Doyle took some real Mormon history and names but everything else came from his own imagination.