Let's divide your sentence into two parts, linked by despite. We'll turn the first part into a sentence by adding a predicate:
A. Mounting signs of danger [ existed ].
B. Few countries or companies had taken the possibility of a pandemic seriously, and there was little interest in developing a vaccine.
The overall structure of the sentence is B, even though A is true. Because A is true, we'd expect B not to be true. But B is true, despite A!
Let's try rephrasing mounting signs of danger as an independent and complete sentence:
Over time, there was more and more evidence that it was dangerous.
Based on this sentence, we'd expect the following reactions:
- Countries and companies should take the possibility of a pandemic seriously.
- There should be interest in developing a vaccine.
But we don't get either of those reactions. We get the opposite. So, we say that we got the opposite despite the mounting signs of danger.