The English carol also may be found at "Hymns and Carols of Christmas," on the internet, a large and useful website.
In addition to the Bramley and Stainer text and music, the music and text as arranged by Martin Shaw and Percy Dearmer are also included.
www.hymnsandcarolsofchristmas.com/Hymns_and_Carols/jacobs_ladder.htm

The African-Americal spiritual and the English carol are both inspired by the same biblical story, but I doubt that they are related other than by source.

The Mudcat DT has only a poor version of the spiritual with a verse by Seeger. The spiritual first appeared in Allan's "Slave Songs of the United States, 1867. Better versions and discussion are given in thread 131 and others.
Thread 131: Jacob's Ladder
Also see Spirituals Permathread.