Non-standard British use of possessive "me"

Native North American speaker here. It's fairly common in certain British dialects to substitute "me" for "my" (Shiver me timbers) in informal speech.

My impression is that some speakers mix the two.

What are the descriptive rules for selecting between the two variants?

Specifically, I'm asking about usage of

A non-standard variant of my (particularly in British dialects) is me. (This may have its origins in the fact that in Middle English my before a consonant was pronounced [mi:], like modern English me, (while me was [me:], similar to modern may) and this was shortened to [mi] or [mɪ], as the pronouns he and we are nowadays; [hi wɒz] he was; versus [ɪt wɒz hi:] it was he. As this vowel was short, it was not subject to the Great Vowel Shift, and so emerged in modern English unchanged.)

(https://en.wikipedia.org/wiki/English_personal_pronouns#Archaic_and_non-standard)

As an example, here is an excerpt from the character Daisy in Downton Abbey:

DAISY
Someone walked over me grave.

(https://scriptline.livejournal.com/43860.html)

Someone please supply a video clip of this speech pattern. I'm sure it can be found, but I can't.

This is a complicated matter. There exists an unstressed form of my which, because of normal vowel reduction of unstressed syllables, is variously pronounced [mi ~ mɪ ~ mɨ ~ mə] without the normal long diphthong [mɑɪ] which you’re used to hearing.

This reduced pronunciation is often heard in the north of Britain today, but it is not strictly limited to that region; this unstressed form has existed in English as long as the language has existed. Its use has been stigmatized, but it is not wrong and it’s certainly common enough.

What arguably is wrong is to misleadingly spell it me as though it were the wrong word. It isn’t: it’s the right word, but people are writing it wrong. They’re using what we call a “spelling-pronunciation” to try to use non-standard spelling to represent dialect speech.

Wikipedia calls this a “pronunciation respelling”:

A pronunciation respelling is a regular phonetic respelling of a word that does have a standard spelling, so as to indicate the pronunciation. Pronunciation respellings are sometimes seen in dictionaries.

It’s like h-dropping in non-tonic syllables. Just because someone says “Give ’er another one” without an [h] doesn’t mean you should normally write her that way.