According to this thread, the OED speculates on its origins inconclusively.
Perhaps so named because English players introduced the technique to the U.S. (but see quot. 1959).
Quot. 1959 being:
1959 Sunday Times 5 Apr. 4/5 The billiard term ‘putting on the english’, which Atticus states is current parlance in American bowling circles. The story goes that an enterprising gentleman from these shores travelled to the United States during the latter part of the last century and impressed the Americans with a demonstration of the effect of ‘side’ on pool or billiard balls. His name was English.
According to Wikipedia, the term is chiefly American and not used by the British. My own guess (roughly in line with the OED), then, is that it comes from snooker being a British game, so it is likely that either the technique originated in England and was associated with English players for a time.
Alternatively, it could have been that there was simply the idea of the game being English, and so it may be expected that the highest standard of play would be found there. If someone demonstrated an impressive skill, perhaps there was the feeling that they were worthy of being associated with the game's parent country.
Further consideration: After looking into the origins of "English" vs "Body English".
"English" was in the 1913 Webster Dictionary, so we know it was around by then. Apparently "English" has been in use since 1869 (more info from OED):
1869 ‘M. TWAIN’ Innocents Abroad xii. 116 You would infallibly put the ‘English’ on the wrong side of the ball.
1877 Chicago Daily Tribune 30 Sept. 8/1 He saw Richard Grant White..miss one shot badly through putting on too much side. ‘Too much English on that,’ said the spectator.
1898 R. HUGHES Lakerim Athletic Club xv. 242 Eaton would slash the ball with a stiffened wrist, an elbow swing, and a quick, hard jump into the air at the same time, to put the ‘English’ on.
1915 Manitoba Free Press Evening Bull. 31 Dec. 6/4 The average billiard player goes to the extreme in the use of ‘English’.
Etymology Online says the following without any backup:
"spin imparted to a ball" (as in billiards), 1860, from Fr. anglé "angled," which is similar to Anglais "English."
From all this, we can fairly safely say that this sense of the word "English" has been around since before 1900.
As for "Body English", Dictionary.com claims its origin as 1905-1910. A quote from the American Heritage Dictionary of Idioms on answers.com says this:
Movements of the body that express a person's feelings, as in His body English tells us just how tired he is. This expression originated about 1900 in such sports as bowling and ice hockey, where a player tries to influence the path of a ball or puck by moving his body in a particular direction. (It was based on the earlier use of English to mean "spin imparted to a ball.")
Finally, according to Merriam-Webster, the first known use of "Body English" was 1908. Long after it's been known that "English" was in use.
So, as far as I can tell from the limited information available, it seems that "English" preceded "Body English".