The 1840 edition introduced the reading and, which replaced the original resultive subordinate conjunction that. We do not know if this change actually represents Joseph Smith’s minor editing for the 1840 edition or whether it is an error made by the typesetter for that edition. By reference to the printer’s manuscript, the RLDS text restored the that in the 1908 edition. Here the editors for the 1920 LDS edition apparently followed the 1840 edition and thus changed the LDS text.
The resultive that implies that the prosperity of the Gentiles is the direct result of the Spirit of the Lord being upon the Gentiles. The and, of course, could be interpreted this same way. As already noted under 1 Nephi 11:29, the tendency to replace the resultive that with and has occurred a number of times in the text.
Summary: Restore the original resultive that in 1 Nephi 13:15 (“that they did prosper”).