Possible sources
Partridge says it's US and Canada slang from c. 1930, and that Norman Franklin says (1976) the original reference is to ther agricultural muck-spreader, and also mentions the following joke as perhaps valid.
The Online Etymology Dictionary says:
The expression [the shit hits the fan] is related to, and may well derive from, an old joke. A man in a crowded bar needed to defecate but couldn't find a bathroom, so he went upstairs and used a hole in the floor. Returning, he found everyone had gone except the bartender, who was cowering behind the bar. When the man asked what had happened, the bartender replied, 'Where were you when the shit hit the fan?' [Hugh Rawson, "Wicked Words," 1989]
US military in WWII
The phrase was at least part of US military slang during World War II, as euphemistic versions can be found in contemporary books, particularly in US Marines accounts of the war. From 1945's The U. S. Marines on Iwo Jima by Raymond Henri et al.
"The garbage hit the fan on that one," said a captain.
The 1947 Star-Spangled Mikado by Frank Raymond Kelley says:
In December, 1945, to borrow a line from an irreverent song
popular among Americans in Tokyo "the Shinto hit the fan."
The song title is also shown in the 1946 The conqueror comes to tea: Japan under MacArthur by John La Cerda:
1949's The old breed: a history of the First Marine Division in World War II by George McMillan tells us the phrase became so popular it was used as a code for a fight or action:
...
Finally, the first I found actually using shit is also from WWII in The Naked and the Dead, the 1948 novel by Norman Mailer:
This one is actually quite straightforward. It alludes to going out on a branch of a tree. Etymonline says that the figurative sense is from 1897. The Phrase Finder supplies a quote from 1895:
The first uses of it in a figurative sense, with no reference to actual trees or climbing, come from the USA at the end of the 19th century. For example, the Steubenville Daily Herald, October 1895:
[...] If we get the 14 votes of Hamilton we've got 'em out on a limb. All we've got to do then is shake it or saw it off.
Best Answer
As this NGram shows, thumb was once less common than fingers. For some reason people have never been particularly keen on thumbs or finger, but they get used too. As Neil Coffey commented, fingers remains more common in British English.
Green hands do in fact exist too, but sometimes simply alluding to the inexperience of general-purpose unskilled employees. I think thumb wins out in the end because the singular is a bit simpler, and thumbs don't have so many other associations that might confuse the imagery.
My chart combines US and UK usage figures, but looking at the patterns for each country separately makes it clear that US thumb and UK fingers became far more dominant in their respective countries from about the 1940s, with little change in usage figures for whatever word was being used on the other side of the pond. I think this may be a result of increased commercialisation of "domestic horticulture" at the time. This probably led to a proliferation of books, magazines, local clubs, etc., and these by their very nature would be far more likely to encourage standardisation around an easily-recognised term.
Thus it may be little more than a fluke that the US standardised on a different term - if neither was particularly well known before, either would be happily embraced by commercial and social networking interests. But standardisation would be very much encouraged thereafter, whichever form happened to have taken the early lead.
Obviously a gardener spends a lot of time handling plant material, much of which is green and might therefore stain one's digits. Since the metaphorical usage is so transparent, it could be re-coined many times. It would normally be understood on first hearing, and often passed on by the hearer, so the expression could have had multiple opportunities to both germinate and thrive.