There’s a Monty Python sketch where it’s pronounced that way - it’s the one where a guy goes to a funeral home and the funeral home guy convinces him that he should eat his dead mother with a bit of broccol-eye and stuffing.
Ireland checking in and it is pronounced “brocollee” here.
Ha, ha, I didn’t know they had that there too. It’s a “true fact” that Catholics in Northern Ireland say “Haitch” while Protestants say “aitch”. 
Were it not for the Python reference, I’d be convinced it was unique to Australia at this point.
It as actually pronounced Broccol-eye as it is the plural of Brocoll (or Broccolo) which are the single florets. Delicious chocolate dipped believe it or not.
I remember having a bit of fun with folks when I moved from Oregon to Texas, and I tried to convince a few people that in Oregon we pronounce it bro-CHO-lee (like cho as in chosen).
But no, I’ve never heard it pronounced anything other than brock-lee or if I did maybe a 3 syllable version that sounds very similar to brock-lee.
My linguistic background is that my father was from upstate NY, as were both his parents, and my mother was from South Miami, father from Cincinnati, mother from Augusta KY.
I pronounce it ‘BRA-gul-Lee.’ With the ‘gul’ really having a schwa in it, like the second syllable of ‘gargle.’ I just find it hard to go from a voiced syllable to an unvoiced one, so ‘broccoli’ becomes ‘broggoli.’
The brocco-lye pronunciation reminds me that although it started in jest, it seems everyone in my family now says asparagi (pronounced ass-par-i-guy). That is, of course, unless they’re going to limit themselves to one piece 
(and this began decades before Tyler and Cameron Winklevoss)
I say /ˈbɹɑk.li/
I’ve heard some others say /ˈbɹɑ.kə.li/, but my two-syllable pronunciation seems more common in the US, unless you ASK people how they pronounce the word, then they affect a third syllable.
Just a friendly tip: there’s no reason you can’t just use /r/ in phonemic transcriptions for English. Just like you didn’t use /ɒ/.
And, yes, asking people how they pronounce something is prone to error. When answering that question, be sure to wait a bit so you aren’t thinking about the pronunciation, and then use it in a sentence you use a lot.
I didn’t use /ɒ/ because there is no rounding. /ɑ/ (not /a/) seemed a better match.
brok-kuh-li
with the middle “u” devoiced so it sounds like 2 syllables
[Newman]Evil weed![/Newman]
Only, if you’re going to treat it as an Italian word (which is what you’re doing there), in Italian they don’t end it with that “eye/ai”, it’s “ee”.
BRAHK-u-lee. Definitely 3 syllables.