Here is what I have been able to find so far: Her name was Emily Johnson. She married William Cogswell when she was in her early thirties, and was a childless widow by the time she was forty.
From North Platte and It's Associations (1910) by A. R. Adamson:
It is worthy of remark that the first Sunday School in North Platte was held in the log school house. The late Mrs. E. J. Cogswell of blessed memory, came to North Platte in 1868 intent upon missionary work. She was a Unitarian, but no sectarian, and willingly co-operated with people of all shades of belief for the moral good of the community. Near the close of that year, aided by Mr. M. C. Keith, Mrs. A. J. Miller and Mrs. Kramph, she had the school room arranged for the reception of scholars to form a Sunday School class, but to the vexation of these excellent women, only three children attended. Mrs. Cogswell, however, was not easily discouraged, and visiting every family in town, in which there were children, she solicited their attendance and was rewarded by having quite a number of scholars. This school was organized as Union Sunday School, and continued for many years.
Up to 1873, North Platte had no hall or suitable place in which to hold meetings or entertainments, and the few Unitarians who had gathered round Mrs. E. J. Cogswell, a missionary of the faith, concluded to erect a building that would serve for a public hall and place of worship. This they did at a cost of $3,300, and it became known as the Unitarian hall. It still stands, battered and weatherworn at the corner of West Fourth and Locust streets, and has passed through many vicissitudes. Unitarianism was never popular in North Platte, and the consequence was, that adherents were few, and funds scant. The American Unitarian association gave liberal financial aid, and sent several pastors in an endeavor to establish a church, but limited audiences and an uncertain salary were not encouraging and none of them remind long. This small body of Christians struggled along for years, some times with a pastor but more often without, until it became almost extinct. Archibald R. Adamson endeavored to rally local Unitarians who had become indifferent, and succeeded in keeping a congregation together for a lengthened period, but he was the last to conduct services in the hall under the Unitarian banner, for dissension caused disruption, and in 1902 the property got into the hands of a very few who sold it and appropriated the money. It was by Mrs. Cogswell’s unwearied zeal that money was raised to pay for the building, and it is questionable if they who profited by the sale ever contributed a cent. The parties in that deal will doubtless feel small when they meet Mrs. Cogswell “in the sweet by and by”, for it was a poor requital for her devotion and labor.
As already stated, Mrs. E. J. Cogswell came to North Platte in 1868 and organized the first Sunday school. She afterwards engaged in teaching and missionary work, and held religious meetings before there were any resident ministers. She also taught music and singing, and performed funeral services in the absence of a clergyman, and was first and foremost in all enterprises for the improvement of the people. Many friends in the east were interested in her work, and contributed books for her Sunday school, money for the support of the church, and clothing for destitute families. She was always planning to help the unfortunate and suffering, and ready to render service to others. Owing to failing health, she returned to her early home in Lexington, Massachusetts, and after two years of feebleness, died on the 23rd of July, 1897. Her devotion to the Unitarian faith continued to the last, and it is [to] be regretted that the latter days of her life were embittered by the knowledge that her work at North Platte was a failure.