Why is Cæsar pronounced “seezer” and not “sayzer” or “sahzer”?

Traditional English pronunciation of Latin – Wikipedia

One of the characteristic features of Anglo-Latin is that the diphthongs æ and œ merged with e. This is fully represented in the American spelling of Latin loanwords, though the simplified spelling is not consistently applied:

æon and eon, æther and ether, amœba and ameba, anæmia and anemia, anæsthesia and anesthesia, cæsura and cesura, chamæleon and chameleon, dæmon and demon, diæresis and dieresis, encyclopædia and encyclopedia, fæces and feces, fœtus and fetus, hyæna and hyena, prætor and pretor

In particular, names were not respelled. So Cæsar was pronouned in Anglo-Latin Cesar—even if it wasn’t spelled Cesar.

When the Great English Vowel shift came to town, the long e ended up changing pronunciation to /iː/, just as it did in English proper. So Classical Latin kajsar > cajsar > tʃajzar > sajzar > Middle Anglo-Latin seːzar > Modern Anglo-Latin siːzar.