Actually, Mel Gibson is American. He was born in Peekskill, NY. His mother was Australian and they moved to New South Wales when he was around 11 or 12 (I think. I can't find an accurate source right this second). His acting career began in Australia and he picked up an Australian accent for the Mad Max movies and Tim, but seemed to lose it pretty quickly when he re-settled in the US and moved his career to Hollywood.
I suspect that at the the time of the American Revolution, all the colonists/revolutionaries/patriots (whatever you want to call them) all spoke English with a decidedly British accent. I doubt that southerners sounded southern and "y'all" probably hadn't been invented.
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Off topic, but I think that's an intriguing comment/point. So when did being southern get invented and when did y'all come about, I wonder. Oh, how long ago American history class was for me. I was thinking it probably started in the deep south (which I believe is officially Louisiana, Mississippi, Arkansas, and Alabama?) and worked it's way north...
For on topic, I guess the closest "Southern" character Heath played was probably Sonny. (Where did that movie take place?) I don't remember thinking anything particular about his accent. (I really liked Steel Magnolias when I first saw it, but now I can't hardly stand to listen to Julia Roberts and her raging Southern voice.)