I will consider only extant pronunciations, without a compounding effect, using the Iraqi pronunciation as a starting base. See the bottom of this answer.
Consonants:
ע* and הᵛ would merge into א, making them all a glottal stop [ʔ].
בֿ would re-merge with בּ^, along with פּ^^, making all three into [b].
ג̄ would re-merge with ג.*ᵛ
ק would merge with כּ*ᵛ, into [k].
ד would re-merge with דּ*ᵛˣ.
ו would shift to [v].
ח and כֿ would merge to [χ]ᵛ.
ט would shift to [t]*ᵛ.
צᵒ and ת* would shift to [s], as would שׁ**, thus merging them all with ס and שׂ.
ז would remain [z]; י would remain [j]; ל would remain [l]; מ would remain [m]; and נ would remain [n]. ר would remain some sort of R sound.
That’s fifteen consonant phonemes.
Regarding vowels: If you include common but incorrect pronunciations, then the vowel pairs Kamatz/Patach, Tzereh/Segol, Chiriq Maleh/Khaser, Cholam/Kamatz-Kattan, Shuruq/Qubbutz, would mergeᵛ. (I will henceforth refer to each now-merged vowel pair by the first vowel in them).
Tzereh and Cholam would merge into /ey/.**
Shuruq would merge with Chiriq, into [i].**
Kamatz would remain phonemic (at least, sometimes).
(Shva would either become universally silent and thus lose relevance, or merge with Tzereh/Cholam to /ey/. Either way, Shva’s usually not phonemic in the first place, even when pronounced).
That’s three vowel phonemes.
Conclusion: 15 consonant phonemes and 3 vowel phonemes; 18 phonemes in total.
You can switch up the list a bit—for example, you can say that ק might merge with א instead of with כּᶦ, or that בֿ might merge with ו instead of with בּ*—but you’d arrive at the same minimum number (unless you know of some obscure tradition that isn’t recorded on the internet).
I am 99% sure that such a pronunciation would not be halachically valid, due to dozens of sources, (a sample of which are brought in the accepted answer). I am also 99% sure that some Rabbis would disagree. Either way, you should definitely not promote this pronunciation. It can barely even be considered Hebrew.
By the way, if you count ancient pronunciations, I can make the list even shorter.
I can also shorten it if you consider a compounding effect—for example, since ו shifted to V and is identical to ב, BOTH could then merge with בּ.
Please comment if either one is the case.
*Ashkenazi
**A variety of Yeshivish/Chassidish
^Mizrahi (non-Arabic)
^^Arabic Mizrahi
ˣNorth African
ᶦYemenite
ᵒSephardi
ᵛIsraeli