1. She'll be coming round the mountain when she comes.
2. She'll be coming around the mountain when she comes.
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In the song, 'round' is used in #1. In British English, 'round is used with action verbs. However, what about in American English. It seems that in American English 'around' is
used instead of 'round.' Do you use 'around' as in #2, in this case, all the time?
In American English, as I was taught and have said and written, "around" is an adverb or preposition, and "round" is a shape (an adjective or maybe a noun).
In the song you're referring to, though, I wonder if the lyrics were written like this:
She'll be coming 'round the mountain when she comes ...