The pronunciation of Latin words and phrases in an English-speaking context is quite complicated, because three separate traditions clash, making it difficult to be consistent. John Wells has been writing about this recently, here, here and here.
First there is the traditional treatment, which arises from the way in which English pronunciation has changed over the last thousand years. The short vowels of Middle English /æ, e, ɪ, ɒ/ have barely changed in stressed position, and as these were probably quite close in quality to the similarly written vowels in Latin, there was no problem, and the English sound was carried over to the Latin words (e.g. etcetera: /etˈsetərə/, quod erat demonstrandum /kwɒd ˈeræt demənˈstrændəm/. Middle English short /u/, on the other hand, has split (in many forms of English) into /ʊ/ and /ʌ/. Generally speaking, short Latin /u/ has become English /ʌ/: cum (used to join two separate village names together when the two merge in some sense ): Stow Cum Quy /ˈstəʊkʌmˈkwaɪ/. The unstressed short vowels generally became schwa, as in the above examples, without anyone noticing particularly.
The long vowels of Middle English have, on the other hand, changed considerably. They were originally just as easily transferred to Latin words and names that also had long vowels, such as mater, regius, Dido. However, with the Great Vowel Shift, Middle English /ˈmaːtər/ became Early Modern English /ˈmeːtər/ and then /ˈmeɪtər/, /ˈreːdʒɪəs/ became /ˈriːdʒɪəs/ and /ˈdiːdoː/ became /ˈdaɪdoʊ/.
The quantities of Classical Latin were not always observed: some long vowels became short, particularly in antepenultimate stressed syllables (so stamen, /ˈsteɪmən/ acquired the plural /ˈstæmɪnə/), while some short vowels were lengthened (via, /vɪə/ became first /ˈviːə/ and then /ˈvaɪə/.
The Latin diphthongs, written AE, AU, OE, had already in Old English times become /eː, oː, eː/ in most Latin dialects, and these pronunciations, as taught and spoken by medieval monks, were taken over into English. They then developed like the other long vowels.
As for the consonants, Latin C before the front close vowels, whether long or short, became /s/, G became /dʒ/ in the same contexts, consonantal I (alternatively written J) also became /dʒ/ before another vowel, and consonantal U (alternatively written V) became /v/ rather than /w/.
This is the explanation for the pronunciation in modern English of Latin names such as Julius Caesar /ˈdʒuːlɪəs ˈsiːzər/, Cicero /ˈsɪsərəʊ/, Catullus /kəˈtʌləs/, and legal phrases like sub judice /sʌb ˈdʒuːdɪsi/.