Forgive me if I use IPA notation. As a non-native speaker of English, I still have some difficulty with English vowels and don't really feel comfortable using English-based systems as Webster's
- In Classical reconstructed pronunciation, it would be ['la.ti.ũ:]. Germans and classicists prefer this one. Note that this is how Romans most likely pronounced it, since there is only indirect (yet quite convincing) evidence. If you are not familiar with IPA, ũ means oo as in goose, but nasalized.
- In Ecclesiastical pronunciation, it is ['la.tsi.um]: ti+vowel not preceded by s becomes tsi. This one was settled down in the 19th century and is well-documented. Although it is most likely not the original way to pronounce Classical Latin, it has ancient roots and enjoys some official status. It is mostly based in how Italian Latinists and churchmen pronounced Latin at the time, and is consistent with what it likely sounded in medieval times.
In both cases: 1) unlike English, t is not aspirated (think of the sound of t in Spanish or Italian). 2) i and u do not form a diphthong, hence they are in separate syllables. 3) since the i is short, the stress goes in the third-to-last syllable (i.e., the first one in this case) 4) a as in father, i as ee in bee, u as oo in boot, but all shorter.
Regarding English pronunciation, there is an intricate set of rules to pronounce Latin in an academic context. It is basically the effect of the same sound changes that affected English plus some academic corrections along history. As a consequence, the very set of rules also changed with time. Following it, you apparently get (thanks Sumelic):
- (Trad. English pronunciation:) ['leɪ.ʃəm], admitting some variants, it seems, like ['leɪ.ʃɨəm], which is pretty close to (if not the same as) what MW says. As Nathaniel suggests, this is how most English-speaking Latin scholars pronounced Latin before the time when Reconstructed and Ecclesiastical pronunciations took force and pushed most local variants close to extinction.
The fact that MW offers a different pronunciation may obey the fact that English evolves with use (rather than what academics say), and the current pronunciation of original Latin words may have departed from what scholars say. For what it's worth, Dictionary.com offers the following one: [ley-shee-uh m], which -correct me if I am wrong- is (almost) consistent with MW's.