It’s actually more complex than you would have expected:
Starting with the US Revolutionary War there was very little difference between US English and British English.
In England proper local usage of English began to evolve what we now think of as an “English” accent when the general population began to mimic how the aristocratic class sounded.
In the US local English began to change under the influence of successive waves of immigrants leading to it’s own distinctive style.
From what I’ve read on the subject modern US English, changed though it has over time, is actually closer to the common English pronunciation of the 18th century than modern British English is.