Well, interesting. Just off the top of my head, and trying to be fairly certain I'm identifying songs correctly, songs that survived in tradition - or are at any rate still sung today (not always as "folk" songs) - include:

BLACK-EY'D SUSAN
THE STORM
THE ARETHUSA
THE DEATH OF ADMIRAL BENBOW
THE LASS OF RICHMOND HILL
TO ANACREON IN HEAVEN
RULE BRITANNIA
YE GENTLEMEN OF ENGLAND