Linguistic comparison is a useful tool in this case. It's the same process used to reconstruct unattested proto-languages. For an idea of its accuracy, when used to construct proto-Basque (that is, the Basque of the Roman era), it proved quite fruitful. Later, tablets would be found in Southern France dating to this period written in this language, and the reconstruction did, indeed, fit. Phonology can generally be deduced by comparing modern variants, as what we speak now didn't come out of nowhere.

By comparing and contrasting historical texts, things like phonology can be deduced. The reason this works is because many of these texts predate standardized spelling, and thus the people writing them are basing their spelling entirely off of how they speak rather than following set guidelines.

Just as well, grammarians existed in the past. One of the best sources for identifying various dialects of Latin within the Roman Empire is a text from one such grammarian, where he, in detail, explains everything wrong with the way the Iberians speak Latin and how to speak "proper" Classical Latin. Other grammarians of the day tell of how the people of modern Provence spoke Latin very well compared to other "foreigners". Some tell when the Gauls started speaking Latin and how quickly their language became so distinct that the Latin of the church was no longer sufficient.

The way that other peoples transcribe certain words also helps this. One reason we can tell that Classical Latin used V to transcribe "U" is not only from context and modern Latin, but because instances of V in Classical Latin are written with upsilon when Greeks attempt to transliterate.

Through these methods, experts are always hard at work uncovering the phonology of peoples past. I hope this answers your questions sufficiently.