The closest term would be a term which only recently fell out of use (about 30 years or so). The word was "lunger" - someone who had a bad lung. More specifically, it refered to someone who was suffering from tuberculosis - a disease that had constant coughing as one of its symptoms. Sometimes older people with a cough or that are constantly short of breath, are referred to as "wheez kid" - a corruption of "whiz kid" - somebody who is very smart.

A witty British pun. :) Berkshire is a county in England's south east, but it's pronounced "bark-shur". So if someone is coughing a lot (sounds like barking), then they should run for local government in "Bark"-shire. I think the phrase is still alive in the UK, but outside of there perhaps we (non-Brits) don't have such a close equivalent. We sometimes say "coughing up a lung".