The Jane Austen Centre already offers a comprehensive article on Christmas music in the Origins of Regency Era Christmas Carols in their Online Magazine, which I cannot add to in a meaningful way, and which includes a lovelingly told history of ‘Silent Night.’
After reading the article, view a YouTube video of Gloucestershire Morris men dancing a traditional stick dance to the tune of While Shepherds Watched, one of the carols described in the article.
While Bledington, where this dance originated, is situated in the Cotswolds, one is quickly transported to the 18th and 19th centuries when viewing this dance and listening to the music. I believe the musical instrument accompanying this dance is the harmonium (thank you for the tip, Pixzlee). Historically, the pipe and tabor accompanied this dance, while later in the 19th century, the fiddler replaced the pipe and tabor musician.
Pipe and Tabor