And this is a very good point. My first Latin teacher was a hard-core classicist; this, added to my linguistics background, lends me to use a Latin pronunciation...usually. I still say [ju-li-us si-zar] and others, including often saying [si-se-row]; same with the Spanish or Italian pronunciation, as the situation warrants. But that's in normal, English conversation. If I was teaching a Classical Literature course, I probably would use the Latin pronunciation. And when I'm teaching my student in Spanish, and want to model the original Latin etymon, I use a Latin pronunciation. When singing a song in Latin, I use the Latin pronunciation.
As for [w] or [v] for 'v', I always use [w]. As far as I've read, it's the correct pronunciation; besides, it's historically linguistically correct for the various results in the Romance descendants. But I don't mind being the odd person!