"So, am I right in thinking the phrase simply means 'Old Fashioned', in which case, perhaps to be perfectly correct, we should read & write 'Old Timey Old Time Music', otherwise the term Old Timey could be referring to absolutely any form of old fashioned music?"

No - because "old timey" apparently is a general expression that can apply to virtually anything suggestive of bygone days, but "Old Timey", capitalized, refers to a specific genre of music.

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"the phrase 'old-timey' was a descriptor that could be applied to almost anything. It meant, and means, an object or action from another, older, time, with the occasional addition of implying that it is quintessentially that."

An interesting observation - my mother uses the term "old-fashion(ed)" in that way, with the added implication that the thing or action thus characterized is somehow possessed of greater validity than its pale, faded, modern, ersatz equivalent. So, for instance, once ever two or three winters, we would be blessed with a "real, old-fashion snowstorm". Some young relative might be a "real, old-fashion farmboy". Etc.