No. They didn't pronounce the S as M. The first thing he says is "caesar sancta e urbis romea augustus" (my latin is rusty as fuck, but I think that's what he's saying) and he's pronouncing all the S's normally. The reason he says Publium Quinctilum Varum, is that Latin uses something called declinations (as does German) which do not exist in english. This basically means that the ending of some words (articles, nounds and adjectives) change depending of the "role" the word is playing in the sentence (subject, direct or indirect object, etc). To oversimplify it, its like conjugating other parts of speech aside from verbs. So basically "Publius Quinctilius Varus" is the name of the senator without declinating it. When he says "Publium Quinctilum Varum" this indicates its the direct object of the sentence, as the emperor has named that man the senator.

I hope this makes at least a bit of sense. My Latin is very forgotten but I can assure you at least that no, they did not pronounce the S as M.

Source: I studied two years of latin in college.