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.

As Kate Bunting says, "vey-dee mee-kuh m" represents a pronunciation based on an old "traditional" system for pronouncing Latin. You can see further details in the Wikipedia article "Traditional English pronunciation of Latin".

Neither pronunciation uses Latin sounds

Neither this pronunciation, nor the alternative pronunciation that you suggested ("wah-day may kum") is the same as the pronunciation that scholars think was used in ancient times (the "reconstructed" pronunciation of Latin). A monolingual English speaker would find it difficult to produce the reconstructed pronunciation, because it includes a number of sounds that don't occur in most accents of English. For example, the sound of ē in Latin was almost certainly not the same as the sound that most English speakers use in the word "may": English speakers tend to make this vowel sound into a diphthong [eɪ], but Latin ē was a monophthong (pure vowel sound) [eː], more like the sound in Italian meno. At the end of a word, the letters -um are thought to have represented a nasalized vowel sound, without any final nasal consonant [m]. Putting it all together, the reconstructed Latin pronunciation of vāde mēcum would be something along the lines of [wade meːkũː], although the exact quality of the vowels is not certain (so it's possible that the [a] might be better transcribed as [ɑ], or the [e] as [ɛ], or the [ũ] as [ʊ̃]).

English students of Latin are sometimes taught an approximation of the reconstructed pronunciation that replaces some of the non-English sounds with sounds that English speakers find similar. But I think it's useful to keep in mind that the English sounds used in this "restored" pronunciation are part of Latin pronunciation only for modern English speakers. As far as I know, no Italian, German, or French speaker would ever use the English "may" diphthong sound to pronounce Latin ē. For more on the difference between the English "ay" sound and the IPA monophthong [e], see the following questions: Why is /e/ generally transcribed as 'ay'?, What rules govern uniform mispronounciation of romance languages?, Why is the pronunciation of French loanwords with the ending é botched?

Both pronunciations that you mention use English sounds to represent Latin; the difference lies in how the English sounds are mapped to Latin sounds/letters. The English sound [eɪ] represents Latin a in the "traditional" pronunciation system, and Latin ē (or sometimes ĕ) in the "restored" pronunciation system.

The "traditional" system is used for other terms

The "traditional" pronunciation system described in the linked Wikipedia article used to be more common (although it's not clear to me whether it was ever universal among English speakers: its alternatives also go back fairly far).

This "traditional" pronunciation is still not uncommon in some law Latin expressions: for example, stare decisis pronounced as "stairy de[saɪsɪs]". The traditional pronunciations of caveat, sine qua non, dramatis personae, and carpe diem do in fact exist and have been used, even if you may not have heard them: they are /ˈkeɪviæt/, /ˈsaɪni kweɪ ˈnɒn/, /ˈdræmətɪs pərˈsoʊniː/, and /ˈcarpi ˈdaɪəm/. It's also not that uncommon for the letter a to be pronounced as the [eɪ] ("ay") sound in nouns taken from Latin, such as hiatus, cloaca, meatus, foramen.

There is no general consensus about "how faithful should one be in pronouncing Latin words in English"; different people use different pronunciations, and it's up to you to decide who to imitate or which system to follow.

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.