I counted the "Heym's" and "Heim's" in the data collection of Kellner & Pape, until now published only in a small number as a private printing. The only Heym with an N mentioned is a Johann Nicolaus Heym, documented for 1852 in Suhl. Sorry, but more info is not available, but the time fits your rifle, which is certainly pre-1870. For that period the books by Heer:"Der Neue Stöckl" may provide more info. K&P list several more Suhl Heyms from the 1852 directory: Gottlieb, Jacob, Stephan, Friedrich Wilhelm, Johann Ernst, Friedrich August, Conrad, Wilhelm.
H.J.Fritze: "Suhl - Heimat der Büchsenmacher" has a listing of the Suhl guntrade, prepared by the US Army after they took Suhl, dated April 13, 1945. It still has two competing Heym "factories":
"Friedrich Wilhelm Heym" at Schillingstr 7 and Mauerstr. 3. This is the predecessor of the today Heym GmbH, Gleichamberg. Founded 1865 by the brothers Christoph and Adolf Heym. Since 1937 "Friedrich Wilhelm Heym".
And "Gebrüder Heym", Schlageterstr. 43, founded 1913 by Richard and Bruno Heym.
In the 1930s there were additional independent Heyms: Max Heym, Drusselstr.31, and Wilhelm Heym, who made the Heym Heeren action falling block rifles. How all these Heyms and Heims (Spelling often changed) were related is beyond my knowledge.