I did the first part of the White Phial quest where I brought the broken phial to the old dude, who basically said "thanks for nothing" and died. My understanding is that a few days later a messenger is supposed to find me and tell me that the old dude's apprentice figured out how to fix the phial. However it's now been a number of days (way more than 3), and no messenger has contacted me. Is this a bug, or should I just wait longer for it to happen?

You need to have the 'Throat of the World' part of the main quest completed and then 72 hours must have passed since you finsihed the first part.