What's the common Chinese term for a chest cold? I can find 支气管炎 for "bronchitis" but I wonder if that's a bit too technical or medical or whether it's common in day-to-day conversation.

In English "bronchitis" is a bit technical so more often people would say "chest cold".

In case it's not clear, this is the kind of cold/flu where you cough up phlegm but don't necessarily have a runny or blocked nose, which are the symptoms of a "head cold".

I can also find the term 胸闷, but I'm guessing that might be used for much more serious chest pains?
气管炎 (tracheitis) and 支气管炎 (bronchitis) are all common words in Chinese. These words are so often used in everyday jokes that they even sounds like everyday slangs.

气管炎 (tracheitis) is pronounced like 妻管严 (the wife controls strictly) in Chinese, so it is often used in everyday jokes for a man who has a bossy wife.