From medieval times up to the early 19th century, every British town and city of any note had a band of waites (modern spelling waits). Town waits were originally the watchmen who patrolled during the night, using a musical instrument to show they were on duty and to mark the hours. As the waits added more instruments and became more proficient at playing them, they developed into bands attired in colorful liveries, adding dignity to ceremonial occasions.
Past Three a Clock is an English Christmas carol written by George Ratcliffe Woodward (1848-1934) based on the traditional tune, London Waits, published by Playford in his Supplement to the Third Edition of the Dancing Master (1657). This tune is now known as the Waits Carol, not to be confused with the Waits Song.
No count-off. Listen for the three triangle notes.