First of all, birds do not have what humans call a “cold”. For birds, it can be a Chlamydial organism, a Mycoplasma organism, or bacterial infection (many types of bacteria can cause respiratory disease).
Wild finches get Chlamydia psittaci which clogs up their sinuses and upper respiratory tract, making it difficult if not impossible to eat. There has also been Mycoplasma infections which do much the same thing.