Although I can't answer the question for the West, I can give the specific example of the mesaharati in Islamic cultures.

The mesaharati was a man whose job it was to, during the month of Ramadan particularly, wake up others for the suhur, or pre-dawn meal. This is extremely important during Ramadan as fasting begins after the sun has risen. So, the mesaharati would walk through the neighborhood calling out "Wake up, oh sleepers! Wake up and praise God!". He would also typically bang a drum or play another instrument to increase his noise, as can be seen in the picture that I linked to above.

This tradition started with Bilal bin Rabah, a freed slave of Muhammad and first muezzin (the man who performs the call to prayer) being charged to wake the Muslims for the Ramadan prayers and meals. This tradition is continued into the present day, even with the advent of alarm clocks, as an important tradition in many Islamic countries.

If you're interested in hearing what this may have sounded like, one of my favorite examples of a mesaharati's song is Napoleon by Hamza Namira. If you listen, the beginning will have the standard drum beat followed by the call to awaken.

I tried to find some source about who woke the mesaharati but unfortunately couldn't find anything. I know this was a rather specific example, but it shows that in some cases it was just a man's job.

In the industrial period, a knocker-upper would do this. This was a man who was employed to go around and tap on the windows to wake up employees. To ensure that the knocker-upper himself was awake in time, he would himself rely on.. a knocker-upper (a shift arrangement, I should think).

Sorry I don't have a better source, this is one of things I have heard of from several different places.