There are many hymns that are based on folk tunes - one could argue that all hymns are based on folk tunes!
Surely too many to list. I am most familiar with the blue (1990) Presbyterian Hymnal, and a few representative examples that are familiar to my congregation and work well with guitar or autoharp:

"Though I May Speak" is beautifully set to "Waly Waly" (or "The Water is Wide")

The Evening Hymn "Day is Done" is set to the tune of "All Thru the Night"

"Simple Gifts" as mentioned earlier may be classified as folk, and is the basis for the folk hymn from the 60s "Lord of The Dance"

The early American hymns "How Firm a Foundation" and "Wondrous Love" are beautiful and great fun to sing together.

For me, there is no more beautiful or haunting song in any genre than "O Come O Come Emmanuel" - it gives me chills every time.

Many of the Spanish hymns in recent hymnals are set to folks tunes - I am thinking of particularly "Canto de Esperanza" and "Pues Si Vivimos"

We tend to sing a diversity of music - different congregations have different tolerance levels, musically. Probably if my congregation had its druthers, it would all be southern gospel. But that's another story...
You might also want to check out the work of John Bell and the "Wild Goose Worship Group" from the community on the Island of Iona. Their worship music is wonderfully written and often based on British, Scottish or Irish ballad tunes.
A link is here

Hope it helps,

Bo