Do we have any idea how quickly the American colonists (specifically those British colonists living in what would later become the United States, but I'd be curious about French and Spanish colonists in their respective areas as well) developed an accent that was distinctly non-European? What is the earliest mention of someone "sounding American," or speaking in a way that they clearly came from the colonies?
It seems that the popular distinction between American and British speech occurred at least as early as 1735 when Francis Moore observed:
It stands upon the flat of a Hill; the Bank of the River (which they in barbarous English call a bluff) is steep
The earliest mention of a specific American (non-European) accent I can find is from 1783. More specifically the accent described seems to be Virginian.
"...a clergyman of Virginia assured me very seriously, that the English of that province was the best in the world; and assigned the same reason in favour of the Virginian pronunciation. /.../
It is true, the North-American English accent is not so animated as that of Middlesex, and the adjoining counties; but it it very perceptible notwithstanding." James Beattie, Dissertations Moral and Critical.London 1783.
This answer is mostly a warning that some reported "American Accent"s may have been nothing of the sort.
The main thesis of the book Albion's Seed is that there were actually 4 basic English accent groups that evolved into the various American English accents. East Anglian, Wessex, North Midlands, and Border English evolved into the American New England, Southern, Midland, and South Midland/Appalachian accents respectively.
Given this, an early visitor to the USA might remark upon an "accent" that was neither universal to the country, nor significantly different from that of the source area in England.
For my money, the best demarcation line is the publication of Noah Webster's American Dictionary of the English Language. This was the first major work that defined the American version of the language to be a different dialect of English, worthy of its own local authorities. Many if not most of USA's unique spellings (eg: "center" instead of "centre") can be traced back to Mr. Webster.