I can't vouch for British or other varieties of English, but to me, "My ladies" sounds not only stilted, but a bit creepy. (If the speaker is a pimp, or maybe 80, then maybe?) If the speaker is, say, a hotel worker, they would say "Ladies this way," not "My ladies," because "My.." would be presumptuous. Note that the English equivalent of "Meine Damen und Herren..." is "Ladies and gentlemen." We never say "My ladies and gentleman."
"My dear lady" or "My dear sir" sound very British, aristocratic old-fashioned. But, they also imply a certain level of familiarity (people who are on your same level of class/status, and you know fairly well.) Whereas "Meine Damen, hierher." sounds to me like something an employee would say to customers of some sort, not to friends (except ironically, as a sort of joke? I'm not a native German speaker, so I may be wrong!) So, "my dear lady" or "my ladies" sounds presumptuous, as if an employee is pretending to be a friend or "equal" of customers.
"My dear lady" is, I think, very rarely used these days. Even in a historical context, it can only be used in certain cases: the "dear" adds a strong emphasis, implying that you are saying something you feel very strongly, or expresing delight, offence, surprise, etc.; here's an example from Boswell's Life of Johnson:
Mrs. Thrale stood to her gun [i.e. argued for her opinion] with great courage, in defence of amorous ditties, which Johnson despised, till he at last silenced her by saying, 'My dear Lady, talk no more of this. Nonsense can be defended but by nonsense.'
You're totally correct. even a Brit would only use the possessive if the adressee was singular ("My Lord/Lady"), and even then only in addressing a senior judge or a member of the aristocracy.
Although, if you're addressing an audience which includes aristocrats, "My Lords, Ladies and Gentlemen" is usual.
And yes, "my dear lady" is something i would expect to hear only from someone of a certain age and/or social status.