I've seen this question asked many times, though of course we might get a solid answer today from a new or different contributor. Previous answers from the FAQ.

Your question is likely based on a misconception, unfortunately. Americans didn't have what we now consider a British dialect. In fact then, as now, there was no single "British accent" or dialect. There are many different accents and dialects in the British Isles as there we back then.

Additionally, it's not the case that the American accent changed while the British one did not. In fact they both changed, and are continuing to change today. All dialects/accents/languages are going through constant change, and each generation of speakers brings about new innovations and new ways of speaking.

Getting to the spirit of your question, there are a couple different factors that are quite significant. The first is that settlers in North America came from many different backgrounds. It's the reason you have a high degree of Scandinavian influence in some American accents and French influence in others. The fact that not every American settler was British means that the accent was already considerably different in many places within a single generation, since the children of those early first-generation immigrants would have grown up in an environment that was very different (linguistically) than their parents.

So the answer to the question of "when did the dialects of the two places begin to diverge" is that it was almost immediate. But it's not the case that in the 1700s Americans sounded like what British English speakers sound like today.

Do have a look at the relevant FAQ section as another commenter has suggested. The question does come up often, and there's more to it than what I've just written. But be sure to approach it without the idea that Americans all sounded like today's Englishmen if you really want a good understanding of the situation.