Like a lot of American 'o's, 'mommy' often sounds (I don't know the IPA for it) more like 'mahmmy' to me (my British ear) - but it's presumably regional. I mean mainly from TV/film, 'valley girl' sort of accent for example: 'oh my gahd he's so haht'.
I'm struggling to think of a single word in an American accent (from American people I have listened to) that has a letter "o" pronounced correctly, as IPA "ɒ". If that's true, and Americans never learn to make that sound until they try to mimic another accent, then shouldn't this be described more as a speech impediment, rather than an accent?
But yes, it's "mahmmy", which is only very subtly different from a British "mummy" - in the American version, there's almost a hint of an "r" at the end of the vowel, whereas the British version has a straight (and short) "u".
I've always thought the difference is more pronounced with the short forms, "mum" vs "mom" (and the red underlining of the latter reminds me of my biases!). I don't think I've heard an American pronounce "mommy" though, so I can't say for certain.
But either pronunciation of "mummy" (embalmed corpse) is how I would address my mother as a child: https://en.wiktionary.org/wiki/mummy