Well, Rana, I may be of a "certain ag" but not of the middle ages! However, as we're both exiled Midlanders I'll forgive you. I'd not heard the term "waiting" but "waits" in the sense of itinerant carol singing with music was certainly still practiced in Brummagem in my youth. This is what the ENCYCLOPÆDIA BRITANNICA online

has to say:
wait

an English town watchman or public musician who sounded the hours of the night. In the later Middle Ages the waits were night watchmen, who sounded horns or even played tunes to mark the hours. In the 15th and 16th centuries waits developed into bands of itinerant musicians who paraded the streets at night at Christmas time. From the early 16th century, London and all the chief boroughs had their corporation waits.

In the 18th and early 19th centuries the custom developed of these ordinary street watchmen serenading householders at Christmas time and calling on the day after Christmas Day to receive a gratuity. When, in 1829, their place as guardians of a city's safety was taken over by the police, private individuals kept up the custom, playing and singing suitable Christmas music.
Tara a bit
RtS