It is poor grammar in either dialect. It should have been “my Mum and I”. My teacher always said that when referring to yourself and another person in the same breath it is a basic courtesy to put the other person first.
"I" is subjective case, and, in this sentence, "I" is the only correct co-subject with "Mum" (or however a Texas kid would refer to his maternal parent).
But, yes, it is good usage and good manners to put the other co-subject (or co-object) first...
Had I been the teacher, I would have stopped the kid and given him an "F" the moment he began his sentence with "Me". And, in an English class, that action would have been perfectly defensible.
OTOH, the teacher's objection to the mere mention of a firearm is totally indefensible, since he assigned the students to select their own topic.