When we say Latin in this context, we don't mean the language of the Roman Empire. We generally mean the language that was and still is sung in music of the Catholic church, and also appears in some secular songs of the middle ages. This is a form of medieval Latin whose pronunciation varies from place to place, but most singers accept certain Italian conventions as standard practice. In order to be consistent we should follow the Italian standard when pronouncing Latin titles, even though we may sometimes notice different pronunciation on some recordings (especially with German choirs).
On the other hand, there are some ensembles with Latin names (e.g. Concentus Musicus Wien, see below), which should always be pronounced the way they would be in their native countries. This makes a difference only for a few consonants.
Standard European rules: a e i o u = ah ay ee oh oo, and additionally y = ee (like Spanish). In casual speech we often pronounce e in the short form eh.
|ae, oe||ay, as in "say"|
|ei||pronounced separately: ay-ee, e.g. dona eis requiem = doh-nah ay-ees ray-kwee-ehm|
Other vowel combinations and diphthongs are basically inuitive, as in Spanish and Italian.
|c [+e,i,oe]||Italian:||ch, as in "church", e.g. coeli = chay-lee|
|German:||ts, like the German z (or soft c)|
|g [+a,u]||g, hard as in "good"|
|g [+e,i]||Italian, English:||j, as in "jello"|
|French:||zh, like French j (or soft g)|
|gn||Italian, French:||ny, as in "canyon", like Spanish ñ, e.g. Agnus Dei = ahn-yoos day-ee|
|English, German:||gn, as in "Agnes"|
|j||y, e.g. Jesu = yay-zoo|
|qu||kw (except usually kv in German)|
|r||r: lightly rolled|
|ti [+vowel]||tsy, e.g. gratius = grah-tsyoos)|
All other consonants are pronounced in the obvious ways.
Latin words with two syllables are stressed on the first. Words with three syllables are stressed on the second if that syllable contains a long vowel; otherwise they're stressed on the first syllable. If you don't know which vowels are supposed to be "long"... go with your instincts, you'll usually be right.
The most common situation in which Latin needs to be pronounced on the radio is when announcing titles of works written for (or inspired by) the Catholic liturgy, of which there are many, and in this case American listeners accept Italian pronunciation as the norm. On the other hand there are ensembles with Latin (or pseudo-Latin) names that do not conform to the Italian rules:
This is Greek, not Latin, but since it's the traditional first part of the text of a Latin mass, you will always see it in a context where Latin is abundant.