bheadh is the third person singular habitual conditional form of the english verb "to be"

I've heard "mbheadh" pronuonce "mayo" as is mayonnaise which makes me want to to pronunce bheadh as "vayo", but I'm thinking this is probably not correct.

Mayo is definitely not correct for mbeadh. Also, you wouldn't get both the "m" and the "h" at the same time at the beginning of the word. It's one or the other depending on the word preceding it.

As for the correct pronunciation of bheadh, something like vyuh, one syllable, would be your standard pronunciation I suppose. I'm more used to pronouncing it with something like a voiced velar fricative /ɣ/ at the end, so vyugh.

First of all, mbheadh is not a word, it should be m-beaḋ (mbeadh in the Caighdeán). The pronunciation "mayo" or "vayo" also doesn't make sense. Using English phonetic transcription, it should be something like "vyekh" in Munster and Galway; "vyef" in Mayo (the county), and something like "vyew" or "vyeoo" in Ulster. Also, in Connaught and Ulster, it is pronounced something like "vyet(ch)" when it comes before pronouns starting with s (i.e. sé, sí, sibh, síad).

You can see here:
https://en.wiktionary.org/wiki/bheadh

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I'm an intermediate speaker of the Corca Dhuibhne dialect of Irish and also have knowledge on the old spelling
Soir gaċ síar, fé ḋeireaḋ thíar

The narrowest transcriptions are in Wagner's Linguistic Atlas and Survey of Irish Dialects.

Wagner transcribes bheadh as /vʹɘəx/, in Cúil Aodha at least. /ɘ/ isn't exactly the /ɛ/ shown in the broader transcriptions in Wiktionary, and is a centralised vowel slightly more front and slightly more closed than /ə/. You can read about this fascinating vowel at https://en.wikipedia.org/wiki/Close-mid ... nded_vowel where there is a good vowel chart. There is also a variant very slightly further back in the mouth, /ɵ/, which is also shown in that chart, so /vʹɵəx/. You can hear this variant in Dónall Bán Ó Céileachair's Irish in the Doegen records in his pronunciation of mbeadh at 00:15 here: https://doegen.ie/LA_1036d1

I always detested the way that Wikipedia and Wiktionary transcribe short e, i, o, u as /ɛ/, /ɪ/, /ɔ/, /ʊ/ instead of just /e/, /i/, /o/, /u/, as those phonemes are typically closer to the actual pronunciation in many dialects, especially those in Munster (Diarmuid Ó Sé talks about how their actually something like /e̞/, /i̞/, /ɔ̝/, u̞/, where the cardinal vowels are lowered when short).

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I'm an intermediate speaker of the Corca Dhuibhne dialect of Irish and also have knowledge on the old spelling
Soir gaċ síar, fé ḋeireaḋ thíar

Just to clarify, I wasn't saying that symbols such as the ones that I previously mentioned should be used in precise phonetic transcription, in which case something like the symbols you mentioned (i.e. ɘ and ɵ, which are ö and o̤ in Wagner's work) and more specific notation such as provided in Gaeilge Chorca Dhuibhne should be given more attention. I just suggest that something like /e̞/, /i̞/, /ɔ̝/, u̞/ should be used instead of /ɛ/, /ɪ/, /ɔ/, /ʊ/ so as to provide a better guide to the learner, even if there is a more complex variation present.

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I'm an intermediate speaker of the Corca Dhuibhne dialect of Irish and also have knowledge on the old spelling
Soir gaċ síar, fé ḋeireaḋ thíar