Note that my question assumes as a given that Star Trek is in an alternate universe which diverged from ours at some point in history.
That seems really, really obvious and undeniable to me.
So the question then becomes: at what point in time did the two universes diverge and have different histories?
This actual question is addressed in the fantastic book Star Trek Chronology by Michael and Denise Okuda.
Unfortunately I can't quite find exactly where it was that they wrote it in the book (I will continue searching for that and update my answer accordingly), but I am sure that they explain that there is a divergence, notably as pointed out in the question the absence of WWIII, the Eugenics Wars, the launch of Voyager 7 and Nomad probes.
This would sit about right with the occurrences, as there seems to be no significant changes to history prior to WWII (the events of corrections to the timeline made by Kirk and Spock in The Guardian of Forever seem to re-establish our timeline). In 1947, the events of Little Green Men occur which are (well we assume anyway :P) in a different timeline. Another aspect is that Khan, assuming he didn't age in the Botany Bay was about in his 40s by the 1990s, meaning he was born prior to TOS slightly after WWII.
If you want to take the viewpoint that any minor discrepancy in the timeline counts as the point of divergence, then you would be looking at Death Wish (VGR) where Quinn takes Voyager back to before the Big Bang - that didn't happen in our universe (that we know of) meaning the two universes diverged before time started from our perspective.
And to save time, I will narrow down the possible time range for the divergence a bit.
Since SOME, but not all, of the Star Trek writers were not very good with history, several historical references from TOS to Enterprise show that the fictional universe of Star Trek was already an alternate universe at least decades before 1966. Possibly centuries or millennia before.
It is certainly useful to know that the divergence point between our universe and the alternate universe of Star Trek is not AFTER 1969 but BEFORE 1966 and so look for it in the correct era of history.
Nobody can find the correct divergence point if they are looking in the wrong era for it.
For those who want specific examples:
What is the episode from the first season of Star Trek: The Original series, the first Star Trek series, that is usually considered the best? "City on the Edge of Forever".
Here is a quote from the transcript of dialog from a Star Trek transcript site:
[1] http://www.chakoteya.net/StarTrek/28.htm [1]:
KIRK: I've seen old photographs of this period. An economic upheaval had occurred.
SPOCK: It was called Depression, circa 1930. Quite barbaric.
And later:
SPOCK: I've slowed down the recording we made from the time vortex.
KIRK: February 23rd, 1936. Six years from now. (reading below the headline FDR confers with slum area 'angel') The President and Edith Keeler conferred for some time today (Then the whole thing goes up in flames.)
KIRK: How bad?
SPOCK: Bad enough.
KIRK: The President and Edith Keeler.
SPOCK: It would seem unlikely, Jim. A few moments ago, I read a 1930 newspaper article.
KIRK: We know her future. Within six years from now, she'll become very important. Nationally famous.
SPOCK: Or Captain, Edith Keeler will die this year. I saw her obituary. Some sort of traffic accident.
Thus the local newspapers from the year they are in are dated to 1930 in the calendar used in the era of Edith Keeler.
Later McCoy talks to Edith Keeler:
MCCOY: The most common question to ask would be, where am I? I don't think I'll ask it.
EDITH: Why not?
MCCOY: The only possible answer would conclusively prove that I'm either unconscious or demented. This looks like old Earth around 1920 or 25.
EDITH: Would you care to try for 30?
MCCOY: I am unconscious, or demented.
This proves that the year is 1930 according to the calendar used by Edith Keeler.
Later:
EDITH: We can talk about that later. I have to go. My young man is taking me to a Clark Gable movie.
MCCOY: A who movie?
EDITH: A Clark Gable. Don't you know?
MCCOY: Well, I know what a movie is, but.
EDITH: That's very strange. You get some rest. I'll see you later.
and Later:
EDITH: If we hurry, maybe we can catch the Clark Gable movie at the Orpheum.
KIRK: What?
EDITH: You know, Doctor McCoy said the same thing.
KIRK: McCoy! Leonard McCoy?
So there are two separate references to a "Clark Gable movie" - despite the fact that several sources have mentioned that Clark Gable did not have any speaking roles in any movies which were released before AD 1931 and AD 1932. It was impossible to have a "Clark Gable movie" in the year AD 1930.
There are only three explanations.
1) Star Trek is in an alternate universe where Clark Gable became a big movie star at least a year or two earlier than in our universe.
2) Star Trek is in an alternate universe where many persons in New York City, and perhaps most of the world, used a different calendar which had a year One which was in year AD Two or year AD Three. Thus "City on the Edge of Forever" took place in AD 1931 or AD 1932 when Clark Gable was starting to be a big star but the people of that era used a calendar in which the year was 1930.
3) Star Trek is in an alternate universe with some combination of the above two.
The previously quoted dialog is sufficient to prove that Star Trek happens in an alternate universe which diverged from our sometime in or before the early 1930s.
"City on the Edge of Forever" has another dialog group which also independently proves that it is in an alternate universe which has diverged from ours in or before the 1940s and 1930s.
SPOCK: This is how history went after McCoy changed it. Here, in the late 1930s. A growing pacifist movement whose influence delayed the United States' entry into the Second World War. While peace negotiations dragged on, Germany had time to complete its heavy-water experiments.
KIRK: Germany. Fascism. Hitler. They won the Second World War.
SPOCK: Because all this lets them develop the A-bomb first. There's no mistake, Captain. Let me run it again. Edith Keeler. Founder of the peace movement.
KIRK: But she was right. Peace was the way.
SPOCK: She was right, but at the wrong time. With the A-bomb, and with their V2 rockets to carry them, Germany captured the world.
KIRK: No.
SPOCK: And all this because McCoy came back and somehow kept her from dying in a street accident as she was meant to. We must stop him, Jim.
This shows that Nazi Germany must have put lot more resources and intelligence into their atomic bomb program than in our universe and/or put a lot more resources into their rocket program.
It is quite possible that an evil neo Nazi time traveler or an extraterrestrial convinced the Nazis, probably back in the 1930s, that guided missiles carrying atomic bombs were the only way to win World War II. Then the Nazi atomic bomb and missile programs would have had more funding and resources and would have been coordinated to build atomic bombs which were small and light enough to be carried by the bigger and longer range rockets they would develop and call V2 weapons.
Thus "City on the Edge of Forever" has two separate and independent proofs that Star Trek is in an alternate universe which diverged from ours at least as early as the early 1930s and at least three decades before the period AD 1966 to AD 1996 which some have supposed the point of divergence was in. And "City on the Edge of Forever" was neither the first nor the last episode of TOS with proof that Star Trek was already in an alternate universe in the 1960s.
IMO, It diverged during WWII, on August 2, 1943, when an obscure B-17 pilot named Gene Roddenberry was killed in a landing accident, when his aircraft overshot the runway at it's base on Espiritu Santo, in the South Pacific. Or, at least, Roddenberry was killed in one universe, where the events he had imagined in rough form for a story he was thinking of writing after the war began to manifest after his death as actual events rather than as fiction. In our universe, Roddenberry survived the accident, and two other men were killed instead. Roddenberry was absolved of any blame for the crash. Later, after the war, he became a writer for television, and the events he had imagined up until August 2, 1943, manifested as a television series. All the events in the parallel universes have been different, to one degree or another, since Roddenberry died/lived in 1943.
If you accept Barbary Hambly's novel, Ishmael, as canon, the divergence is apparently prior to 1860. In that novel, the events around the Mercer Girls episode of Seattle, Washington history (which, in our timeline, occurred in the 1860s) were already altered prior to Kirk, Spock, and McCoy (accidentally?) traveling to that time -- and landing in the timeline of TV series Here Come the Brides. For instance, Asa Mercer was replaced by the Bolt brothers and Henry Yesler by Aaron Stemple (who turned out to be an ancestor of Amanda Grayson, hence of Spock).
Unless time travel in the Trek continuum crosses lines, this firmly places TOS and Brides in the same timeline, which was already diverged from ours at least as far back as 1852 (when the real Henry Yesler arrived in Seattle).
Could it be when Gary Seven was initially plucked from our history around 4000 BC. Perhaps his presence in the universe setup our present reality. Without him, his prodigy would never have an impact on our world. Also, who's to say his presence on his hosting planet before being returned to Earth didn't prevent an eventual invasion of Earth which may still occur in our real timeline.
