Joe, that looks like an 'augmented' version constructed from more than one spiritual or otherwise composed.
The originals do not have those verses. "Go Tell It On the Mountain" was first printed in 1909, "Religious Folk Songs of the Negro," at the time credited to "Hampton Institute and Fenner" (Fenner long dead, Robert R. Moton was Commandant; Dett not in charge until the 1927 edition).
As posted by Masato in '01, there were only two verses and the refrain in that initial printing.
Fisher (Seventy Negro Spirituals, 1926) used the same two verses in his large sheet music edition.

The "African American Heritage Hymnal," 2001, GIA Publications, has these verses (no. 202) -
1
While shepherds kept their watching
O'er silent flocks by night,
Behold, throughout the heavens
There shone a holy light.
2
The shepherds feared and trembled
When lo! above the earth
Rang out the angel chorus
That hailed our Savior's birth.
3
Down in a lowly manger
The humble Christ was born,
And God sent us salvation
That blessed Christmas morn.
Chorus:
Go tell it on the mountain,
Over the hills and everywhere,
Go tell it on the mountain
that Jesus Christ is born.

These verses directly copy those of John Work Sr. and printed in "American Negro Songs and Spirituals," 1940.
"Songs of Zion," 1981, ed. J. Jefferson Cleveland, uses the same three verses from Work.
(These two volumes both prepared by African-American groups).

I have not seen the version of the Utica Jubilee Singers, which may have different verses.