I know that it all depends on the health of the individual, but typically speaking how fast can a common cold progress and develop into pneumonia?
After how many days of enduring chest congestion and coughing up mucus, from what appears to be a typical cold, should one be concerned that it may be pneumonia? Is three days too soon?
First, the definitions: A common cold (or "head cold") is a viral infection of the nose and throat. Pneumonia is an infection of the lungs. Tracheitis and bronchitis can be refered as a "chest cold" but not common cold (WebMD).
Neither a common nor chest cold are necessary steps in developing pneumonia. You can get pneumonia without having a cold. Here's a MedlinePlus article about viral pneumonia that does not even mention runny nose or sneezing, which are typical common cold symptoms.
Or you can get pneumonia at the same time as common cold, or days or weeks before or after it. This is because you do not need to go through a typical scenario of catching cold from someone and then developing pneumonia. You can have viruses, for example, influenza virus, in your nose, mouth or throat as part of "normal flora" without having any symptoms. At the time when your immunity weakens from some reason, the viruses can multiply and cause symptoms, either in your nose/throat (a common cold) or, if you inhale them, in the trachea/bronchi (tracheitis/bronchitis) or in your lungs (pneumonia).
The bottom line is that you may not be able to make a diagnosis from timing but from symptoms, as Carey Gregory mentioned in the comment. In a common or chest cold, you usually have no fever and in pneumonia you usually have it (but not always).
