The Word of the Day for April 7th, 2019 on Dictionary.com is vade mecum, coming from the Latin expression vāde mēcum meaning something like "come along with me." Dictionary.com lists the pronunciation options as,

[vey-dee mee-kuh m, vah-]

It's been some time since I've studied Latin, but a couple of discrepancies are obvious to me here. In Latin, this would be pronounced

wah deɪ meɪ cum/[wah-day may kum]

There's no "v" sound in Latin. That changes to "w". This change is understandable though. Nobody pronounces veni, vidi, vici with the "w" sound. However, the long A and E sounds they're using in the first syllables of both words are also absent in Latin, and, in thinking of some other Latin loan words--caveat, sine qua non, dramatis personae, carpe diem--I don't think I've encountered those vowel sounds before. So, considering that Dictionary.com is somewhat authoritative in providing pronunciation guidance, what are they basing their pronunciation on; and, further, how faithful should one be in pronouncing Latin words in English?

On the Classical vs Traditional pronunciation of Latin words in English

For your term’s pronunciation, its (paywalled) OED entry for vade-mecum lists first the /ˈvɑːdeɪ ˈmeɪkəm/ version with the FATHER vowel for the first stressed syllable and the FACE vowel for the second. Then following that one it has /ˈveɪdi ˈmiːkəm/ pronunciation now showing the FACE vowel in the first word and the FLEECE vowel in the second.

Notice how what had originally been a Latin imperative verb phrase is now uses as a noun in English and the other tongues that use the term, just like we nouned ignoramus from a Latin verb to an English noun.

Etymology: Latin, vāde imperative singular of vādĕre to go + mēcum with me. So French vademecum, Spanish vademecum, Portuguese vademecum (Portuguese also vademeco).

These two very different pronunciations respectively represent the so-called “classical” versus “traditional” pronunciation of Latin words in English, as mentioned in this answer and laboriously detailed in this very long Wikipedia article.

The essential difference is that the first one is far closer to the “classical” pronunciation of Latin. That is, it’s pronounced “as it’s spelled” using the original values of the Latin letters as they’re still used in the vast majority of non-English languages and indeed how they’re used in the International Phonetic Alphabet. It’s how someone who speaks Italian or Spanish or French or Portuguese or Romanian or German or Swedish would expect to pronounce it.

The only phonetic accommodations made are those required by the phonotactics of English pronunciation: Latin /e/ getting the customary non-phonemic off-glide we sometimes write as [eɪ] or [ej], and the characteristic reduction in unstressed syllables centralizing that /u/ to a schwa /ə/. V was just another way of writing U, just as J was just another way of writing I, but in a consonantal use that letter was probably realized as [β̞], as voiced bilabial approximant or fricative that English doesn’t have but which can be still be found in Spanish for intervocalic ‹v› (and ‹b›). That's necessarily been altered to a sound that English does actually have, the /v/ you see there.

The second “traditional” pronunciation would be the one you would expect a native speaker English with no knowledge of how any other language used the Latin letters for vowels. When Old English started spelling things using the Latin alphabet, they quite reasonably used the Latin letters corresponding to those sounds. But then when time mutated nearly all of those, the spelling never changed to match the pronunciation shifts.

That's because the Great Vowel Shift, which changed how English pronounces nearly all words, came about after we started writing words down using the Latin alphabet. So we ended up using the "wrong" values for almost all of these letters from the perspective of the rest of the world. Under the shift, ‹a› becomes /e/, ‹e› becomes /i/, ‹i› becomes the phonemic diphthong /ɑɪ/, and ‹o› often becomes /ɔ/ or even /ɒ/ or /ɑ/. (There are also a few consonant changes, like ‹c› no longer always representing /k/ but sometimes /s/ instead.)

I’ve never personally heard anyone use anything except the classical pronunciation of vade mecum, the way you might expect of an Italian speaker or a Spanish speaker, the one that the OED lists first. This might well be because I’ve also never heard it uttered by someone without any background in a Romance or other continental language, let alone in Latin proper.

It’s probably a markedly “learnèd” term these days, one you would only see in a more scholarly context. The OED places it is its frequency band three, along with such terms as ebullition and prelapsarian, contumacious and argentiferous. So even though you wouldn’t expect to find it in newspapers (apart from The Economist :) neither is it a term that should puzzle educated readers.

English speakers used to have a conventional way of pronouncing Latin, which was baffling to Latin speakers from continental Europe. It survives in expressions like vice versa and via (as in 'by way of'), and in some scientific and legal terminology. See https://www.cambridge.org/core/services/aop-cambridge-core/content/view/A0860C6625BE5A0E45FD58A18797E6FB/S175027051200005Xa.pdf/the-english-pronunciation-of-latin-its-rise-and-fall.pdf

However, my Pocket Oxford Dictionary, 1992 revision, tells you to pronounce vade mecum with the 'ah' sound.