According to the following source the adage The apple doesn't fall far from the tree originated in AmE in the first half of the 19th century:
The first recorded use in the USA was by Ralph Waldo Emerson in 1839, one of America's best known 19th century figures.
But they also add that:
Versions of this proverb can also be found earlier in works written in German and Russian; with some sources saying the expression originates in Asia. (www.bookbrowse.com)
As for its possible German origin, I could find the the following test from 1842 The Exercises for writing German - Page 14 by Johann Gerhard Tiarks - where they state that:
"The apple does not fall far from the stem," is a German proverb.
But what is more interesting, in A Dictionary of the Welsh Language, Explained in English dated 1803 under the term Avall (apple) they quote:
Ni fell zygwyz aval o avall; the apple will not fall far from the tree, (adage).
The Welsh quote is the earliest I could find, but it just seems to make my research more complicated.
So, where does the above adage really come from? Is is ultimately of Asian origin as suggested in one cited source?
Was its earliest usage in AmE as suggested, or does the Welsh dictionary proves otherwise a possible BrE first usage before it crossed the pond?