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I've heard more than a few Indians speak in an accent that very close to Scottish.

Caribbean English accents also remind me of Scottish.

I wonder if there were certain areas of the British Empire that had disproportionate Scottish influence.

Yes there is. Niall ferguson describes it as "the empires [Indian colonial administrations] distinctly celtic tinge" if memory serves me right.

People attempting to do a Welsh accent often make a stereotypical Indian accent.