My child (currently 9 months old) has gotten his fair share of colds so far, and I expect more to come.
Sometimes, he coughs and I can hear the phlegm come up in his throat. Aside from the fact that neither he nor I like the bulb syringe, it seems physically impossible to get to it in time before he just swallows anything he may have coughed up.
Should I be doing anything about this and, if so, what is it?
As far as I can tell, there isn't really anything you can reliably do to directly remove the coughed up phlegm at that age. The bulb syringe or just wiping it out of his mouth with clean gauze, washcloth or similar is the most viable option if there's a large amount of phlegm, but I've found that that benefits of not swallowing it (maybe prevent the baby from vomiting later), are not huge compared to the difficulty of actually accomplishing it and how invasive and obviously unpleasant/disliked it is for the already unhappy sick baby.
For comfort and making it easier to keep his throat clear, you can run a humidifier, hold him in a steamy bathroom with the shower running hot, and offer extra fluids if he's willing to take them (water, breastmilk or formula, some parents offer diluted juice). Keeping his sinuses cleared with saline and the bulb (or other suction solution like the 'vacuum' or NoseFreida, ect), can also help with the overall mucus situation.
There is a technique I have seen used to help break up chest congestion, where you sort of hit the person's back repeatedly. I've seen a physical therapist do this for a patient, and at some point I got permission from my son's doctor to do this for him. However, my son was older than your child. For this idea, and in general, I would encourage you to ask the doctor.
One other note, I've been told that dairy products can thicken mucous. My children's doctor encouraged us to keep the dairy products to a minimum during respiratory infections.
Edit: I looked to see if others agree with my children's doctor about a dairy connection, and apparently the jury is still out on this. I found several articles that called the dairy-mucous connection a myth; here's one that did draw some connection:
For some people, dairy products can cause mucus to thicken up, and that may cause sinus pressure and congestion. (source)
I do want to add that what you heard might be wheezing, which it might be good to have a doctor give a listen to.