This is not a substitution of me for my but a common dialectal pronunciation of my (Northern England and some Australian accents).
According to Wikipedia, other areas of the North have regularised the pronouns in the opposite direction, with meself used instead of myself.
Moreover, the vowel in 'me' (the one you're referring to) is shorter ([mi]) than that of the regular 'me' ([mi:]) and this pronuncation of 'my' is usually used where 'my' is unstressed.
The pronunciation of emphatic/stressed 'my' was [mi:] in Middle English (before the Great Vowel Shift) and its unstressed/unemphatic form was [mi].
The Great Vowel Shift was a vowel change (1400-1700) that shifted almost all the long vowels to diphthongs but it did not affect short vowels. So the pronunciation of emphatic 'my' shifted to [aɪ]* but the short vowel in 'my' [mi] in its unemphatic form remained unaffected in some accents (Northern British, Irish, some Australian accents etc).
*GVS shifted [iː] to [aɪ]. For example, the pronunciation of 'bite' was /bi:t/ before the GVS.
miformy. More idiomatic would be 'The wind was so strong I nearly lostmipants." Though if your accent would usemilike that, it would also transformIintoàlike inlad– DoneWithThis. Aug 16 '20 at 10:27Inotmiformy. – DoneWithThis. Aug 16 '20 at 18:29