Ann: Cough analysis patent granted in China, page-86

As @FredP already noted, the results can be calculated locally on the phone without any connectivity. It might take a few seconds, but certainly less than a minute. This is because the analysis and training work identifies specific acoustic signatures that discriminate between coughs from subjects with the target condition and coughs from subjects without it. Doing the training and analysis takes enormous amounts of data and computing.

But once complete the relevant signatures can be extracted from a recording and the trained algorithm make its decision very quickly.

In fact, as we've discovered, it's integration with medical reporting and practice management systems that is the greater source of complexity with the phone app.