I read that the British accent diverged from the American accent and not the other way around. When did that happen?
To say that the British accent came from America isn't accurate. The two dialects developed independently and it's largely down to how regional disparities in Britain were more prevalent than in the US.
John Hurt Fisher agrees that there was initially no difference in dialects between British and Americans; "In the impressment controversies of the 1790s, naval officers on both sides found it so difficult to tell whether sailors were British or American that the American government considered providing certificates of citizenship". He goes onto write that dialects in both Britain and America were region based, but as Britain began to shift to a more wealth and prestige driven society, the 'Language of London' became the desirable dialect of wealth and prestige.
Michael Montgomery goes onto write that America didn't really have regional dialects early on in the settlement of America, but that they developed as each region (such as New England, and 'the Middle States') began to develop a dialect independently.
He cites two written sources;
Though the inhabitants of the Country are composed of different Nations and different languages, yet it is very remarkable that they in general speak better English than the English do. No County or Colonial dialect is to be distinguished here, except it be the New Englanders, who have a sort of whining cadence that I cannot describe.
I ought perhaps to except [from the universal prevalence of dialect] the United States of America, in which dialect is hardly known; unless some scanty remains of the croaking, gutteral idioms of the Dutch, still observable in New York; the Scotch-Irish, as it used to be called, in some of the back settlers of the Middle States; and the whining, canting drawl brought by some republican, Oliverian and Puritan emigrants from the West of England, and still kept up by their unregenerated descendants of New England - may still be called dialects.
Britain had many more regional dialects firmly ingrained in society as a result of it's long history. American regional dialects therefore took a while to develop as the settlement process went on. The 'proper' English accent is therefore a result of the wealthy looking for a desirable accent to set themselves apart from regional dialects.
Montgomery does go on to write what Americans thought of their regional dialects (after they began to notice them cropping up) in relation to British English and whether it was seen as archaic English, an offshoot of 'Elizabethan English' in isolated areas or a result of the immigrants' individual native regional tongues.