I can try and answer this partially for you as a physician.
Smoking was nowhere near as prevalent before the 1900s as it would be by mid-century, when over half of US adults were regular smokers. The per capita number of cigarettes smoked per year in the US increased from less than 100 in 1900, to over 4000 by the 1960s. This was driven by a combination of the development of mass production technology for cigarettes, as well as marketing.
The implication of this is that many long term health effects of smoking were not well known prior to the early to mid 1900s - there were simply orders of magnitude fewer people smoking.
However, we can infer that some more immediate effects of regular cigarette smoking were widely known to the greater public by the 1930-40s based on documented marketing campaigns. For examples of throat irritation or shortness of breath:
They don’t get your wind” (Camel, 1935), “gentle on my throat” (Lucky Strike, 1937), “play safe with your throat” (Phillip Morris, 1941)
For an example of association with cough:
Not a cough in a carload” (Old Gold) (U.S. Federal Trade Commission [FTC] 1964, p. LBA-5); “we removed from the tobacco harmful corrosive acids (pungent irritants) present in cigarettes manufactured in the old fashioned way”
Chronic cough from smoking doesn’t mean you have COPD or emphysema though, rather that you have chronic bronchitis. Historically, these diseases were much less common than they became in the 1900s, and were mainly caused by pathology other than smoking. Oslers Principles and Practices of Medicine from 1916 barely even mentions emphysema as a disease entity, which contrasts the situation by the modern era where it was the cause of 5-10% of all deaths worldwide.
While physicians and others noticed the rise in cases in chronic bronchitis/COPD/emphysema in the 1900s, it was not immediately clear at the time that cigarette smoking was causative. It was not until the same report in 1964 that suggested smoking may cause lung cancer also mentioned smoking may cause chronic lung disease, that this possibility really became public knowledge.
To get to the point of your question, based on the above, I would suggest the inference that patients may recognize that smoking may cause a cough (as marketing materials about this association suggests this would be widespread knowledge), and they may see their physician and be diagnosed with chronic bronchitis, but that they and their physicians would be unlikely to think that cigarette smoking would be the causative agent of more significant lung disease (like COPD), at least prior to the 1960s.
References:
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3894634/
https://www.ncbi.nlm.nih.gov/books/NBK294310/
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2706597/
https://www.who.int/news-room/fact-sheets/detail/chronic-obstructive-pulmonary-disease-(copd)
https://www.ncbi.nlm.nih.gov/books/NBK53021/
Edit: one further reference that supports the notion that cigarette smokers were known to develop a “smokers cough,” by the early 1900s, but this was felt to be benign.
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3456831/pdf/11524_2006_Article_BF02345669.pdf