The original signed copy of the Declaration was promptly taken to the printer John Dunlap, who made copies which were sent throughout the thirteen states; the original was kept in Congress's archives throughout the war (and is still in the National Archives today.)

Just like /u/Holbac, I've never heard whether any formal copies were sent to England, though they promptly heard about it. According to Todd Andrlik, editor of Journal of the American Revolution, the first public report of the Declaration in Britain was on August 10th, as one "bullet point" in a list of short bits of news:

Advice is received that the Congress resolved upon independence the 4th of July; and, it is said, have declared war against Great Britain in form.