Mine is not dialect for my, rather my is a reduced form of mine. For a while my and mine co-existed like a and an, mine/myn before vowels and my before consonants.
Even after my became the standard form in formal speech, mine sometimes returns before vowels, e.g. "mine eyes have seen the glory ...". So
Nancy Nell and
Ned are examples of rebracketing like adder, apron and newt.