English Folk Music and Christmas carols are closely linked together. Carols like The Holy and the Ivy are many centuries old, based on an oral tradition, just like a number of folk songs. Many of these carols actually have elements of paganism and religiosity mixed together, just like many folk songs. The Holy and the Ivy is a good example of this. If you want to learn more about this carol, go to the Wiki article about it.
Right up until the end of the last century, still continuing in some areas, carol singers used to go around houses and pubs in the UK singing sometimes playing instruments, collecting money, usually for charities. Nowadays the most likely carol singers are actually Salvation Army bands in shopping centres. But a number of contemporary English folk artists have continued the tradition of Christmas and Winter songs with their recordings.