Jazz has become an international language. Euro-jazz is different from American jazz as is Asian, Middle Eastern, Hispanic etc. Today every nation can claim jazz as they do it as their own.

Improvisation is at the root of jazz and is not new. Mozart could do it as well as a New Orleans street musician.

English folk song as the root of jazz? I don't think so which doesn't devalue English folk music. There is a tendency in all music to conform to an idea of its native form and development and deviation from that is not tolerated. This is the case with English folk song, American Bluegrass, child ballad singing or any other rigid ideas about performing such music.

Jazz helps us to realize just how interconnected and fluid music is. All forms of music are amalgams of earlier forms adapted to new times. Too many folk musicians tend to be antiquarian in their nostalgia ignoring that the music that they prefer was at one time new and different. Bach considered the dominant seventh chord dissonant in his day requiring immediate resolution. The flatted-fifth be bop chord would have been burned at the musical stake in those days.

Music has to be understood as different languages for varied cultures and nations.
You put on a different set of ears to listen to Jeannie Robertson or Margaret Barry,
Almeda Riddle, Horton Barker, or Texas Gladden then you do to hear Bird, Monk or Trane, Louis, Tatum, etc., each having its own musical parameters to understanding it.

The origin of the term "blues" may have come from William Shakespeare's "blue devils" but what we familiarly know as the blues is from American south, Mississippi, Alabama, etc.which is not to say that it couldn't be applied to any sad, deep music from Flamenco "conte hondo" to Irish "keening" or the shepherd's horn on the Steppes of Central Asia (see the opening bars of Stravinky's Rites of Spring).

Musical terms are continuing to be redefined such as "folk" or "jazz" as different languages grow and borrow from each other creating new words.