Q. Would it be “the Cherokee Nation” or “the Cherokee nation”?

A. For an answer to this question, we refer you to the indispensable Elements of Indigenous Style, by Gregory Younging (Brush, 2018). Younging’s book is written from a Canadian perspective, but much of its advice related to capitalization applies equally to usage in the United States:

Nation: This term has become widely accepted by Indigenous Peoples to describe separate Indigenous groups as political entities. . . . Nation is usually embedded in the name of a particular Indigenous People, and as such is capitalized—for example, Six Nations of the Grand River, the Métis Nation of Alberta, and Bigstone Cree Nation. (p. 68)

So refer to the Cherokee Nation with a capital N. The entry quoted above goes on to note that lowercase nation would be appropriate in general contexts (as in the plural): “the nations of North America before contact with Europeans” (p. 68). For nation in non-Indigenous contexts, where it is usually lowercase, see this Q&A.