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I found, in the Cambridge dictionary, that the first vowels in American English of the two words August and bother are the same. They are all notated as a /ɑː/.
However, I found in other dictionaries, such as Merriam-webster and Longman, that the first vowels of the two words in American English are not the same.
What exactly is the American English pronunciation in the real world, and what's the problem with the dictionaries?
There is no single American pronunciation of these or of nearly any other English word.
What’s happening is Cambridge has chosen a written phonetic representation that fails to correspond to the pronunciation in their own sound clip. This is unfortunate. It is not true that 333 million Americans cannot pronounce /ɔ/ or /ɒ/, only /ɑ/. You can hear that the American sound clip they provide does not use /ɑ/.
The OED gives UK /ˈɔːɡəst/, US either /ˈɔɡəst/ or /ˈɑɡəst/. The first of those two is the more common pronunciation in North America, and does not differ substantially from the UK pronunciation.
The latter version with its unrounded vowel occurs only in speakers with the LOT–CLOTH split yielding unrounded LOT, the FATHER–BOTHER merger, and the version of the COT–CAUGHT merger that yields only the FATHER vowel, never the CLOTH or THOUGHT vowel.
Younger speakers from southern California may do this, and it is apparently those speakers’ speech that Cambridge chooses to represent in its notation. But their own sound clip belies that position.
See the Wikipedia article on Phonological history of English open back vowels.
You should also look at and listen to the many real pronunciations of daughter, long, and warm at the Sound Comparisons website. Do you see how much variation there is in the real world? A lot!
This should chase away any notion that there is ever just one possible exact pronunciation of English words, particularly those two that you mention here.
The Cambridge Dictionary systematically replaces /ɔː/ with /ɑː/ for American English words. For example, tall and dawn both have /ɑː/, and these are two words that Americans pronounce with /ɔː/ if they pronounce any words with /ɔː/.
This is very characteristic of American speakers who grew up West of the Mississippi river, but there are still a lot of American speakers in the East who distinguish between /ɔː/ and /ɑː/, e.g., who pronounce dawn and don differently.
The Merriam-Webster Learner's Dictionary does the same thing, but in this case, there is some justification for it—if you're learning a language, it's easier if there are fewer phonemes you have to worry about. Their non-learner's dictionaries keep /ɔ/ and /ɑ/ separate.
So why does the Cambridge Dictionary do this? I don't have an answer.
