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Since SOME, but not all, of the Star Trek writers were not very good with history, several historical references from TOS to Enterprise show that the fictional universe of Star Trek was already an alternate universe at least decades before 1966.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:
 http://www.chakoteya.net/StarTrek/28.htm :
KIRK: I've seen old photographs of this period. An economic upheaval had occurred.
SPOCK: It was called Depression, circa 1930. Quite barbaric.
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.
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.
EDITH: If we hurry, maybe we can catch the Clark Gable movie at the Orpheum.
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.
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.