Short answer: An awful lot

Longer answer: Joseph Smith first sent missionaries to England in the late-1830s. These missionaries found so much success that for the next several decades, the vast majority of LDS converts came from Europe, especially England. There were also missionaries sent to Italy, Scandinavia, India, and Australia, over the next few decades.

Mormons established their mission headquarters in Liverpool. They printed a magazine there called the Latter-Day Saints' Millennial Star, which reprinted speeches from church leaders in Utah, and shared news of the missionaries. They made tens of thousands of converts in Europe, mostly from lower-class laborers. These converts were encouraged to gather to "Zion"—Utah Territory.

These converts would save money, often for years, or, later, get a loan from the church's Perpetual Immigration Fund to sail to the United States. Dozens of ships were contracted to bring thousands of immigrants. These were most often sailing ships, though later in the nineteenth century, some were steamers, but they were more expensive and so they were rarely used. Immigrants would land in New York City or New Orleans, sometimes take work to raise more money for railroad travel, then make their way to St. Louis or Omaha. Once again, they would often take work to raise funds to make the trek west, until the transcontinental railroad was finished in 1869.

Sometimes one traces a convert family from London to the United States, only to find that the family ended up abandoning Mormonism and stayed working in St. Louis or New Orleans or New York. Most often, however, these families made their way west to Utah. Once there, they might be assigned to settle in an LDS colony throughout the west—places like St. George or Parowan in southern Utah, or even San Bernardino in California, which was originally founded as a Mormon colony. Scandinavian converts, for example, almost always settled in Sanpete County, Utah.

Utah then, throughout the mid- to late-nineteenth century, was awash in immigrants and accents. Dozens of prominent church leaders spoke with a British accent, a Danish accent, a Swedish accent. George Q. Cannon, one of the best-known Mormons of the nineteenth century, who spent years lobbying Congress for Utah statehood, was British. Anthon H. Lund, a Danish convert who came with his grandmother in the early 1850s, became an apostle and later a member of the First Presidency, Mormonism's highest governing body. John A. Widtsoe, a Norwegian, also became an apostle. Apostles George Teasdale and James E. Talmage were British, Charles Callis was Irish. But as the years went on, most LDS Church leaders were born in the United States and, increasingly in Utah and Idaho, the children and grandchildren of the immigrants who came in the mid-nineteenth century. After the deaths of Widtsoe and Callis in the mid-twentieth century, I don't think any church leaders came from outside the United States until Dieter F. Uchtdorf some fifty years later in 2004.

The LDS Church also made inroads in the Sandwich (today Hawaiian) Islands and to this day Mormonism has large numbers of Polynesian members and converts. Unlike European immigrants, however, these Polynesian converts were never selected for high church office.

By the 1880s, as Mormonism came more and more under scrutiny because of polygamy, Congress moved to restrict immigration by LDS converts. Then, after the Manifesto of 1890 announced that Mormons would stop practicing polygamy (they didn't, but that's a long story for another time), and Utah achieved statehood in 1896, LDS leaders announced that converts in other countries should remain in their homes and build up the church there instead of traveling to Utah. Of course, some members still chose to move to Utah, and students might travel to attend Brigham Young University, or stay with family members, etc.

There are some specific books on Mormonism outside of the U.S., especially in Britain, but anyone interested should start with a general history of the church in the nineteenth century, such as Arrington and Bitton, The Mormon Experience (New York: Knopf, 1979). For more in-depth books, see Allen, Esplin, and Whittaker, Men with a Mission (Salt Lake City: Deseret Book, 1992); Britsch, Unto the Islands of the Sea (Salt Lake City: Deseret Book, 1986).