Indeed, pestilentia does seem to describe any contagious disease as well as actual plague, just like Italian pestilenza, Spanish pestilencia, and Old French pestilence. The Romans didn't think twice about pretty much reducing any epidemic to pestis, plague.
Actually I couldn't find any attestation of something like lebra est pestilentia, which would close the case, but all the dictionaries I checked agree.
However the Romans did distinguish them. Two more generic terms are lues and tabes.
I see the L&S entry for the latter doesn't include this meaning, but this article by Gigliola Maggiulli, in its introduction, confirms this: "pestis competes with broad terms such as pestilentia, lues, tabes, and the generic virus and morbus".
In De Coniuratione Catilinae, Sallust compares the stubborness of the conspirators not to reveal their plot, to an epidemic:
[...] tanta vis morbi atque uti tabes plerosque civium animos invaserat.
[...] such was the violence of the disease that, like an epidemic, had infected the minds of the citizens.
I was about to make an edit to say that adding an adjective meaning "global" would sufficiently characterise a pandemic, when I saw on Twitter that just today, Pope Francis wrote pestilentia universalis for "pandemic".
Skimming through his tweets, it's interesting to note that in these months, he has always uniquely used pestilentia, sometimes coupled with universalis, and never the likes of lues, tabes or even epidemia.