OK, here's the start of the timeline for what I think the STTL's alternate 20th Century might look like. This is the first time I've ever done anything like this! I'm only up to the end of 1945 so far; I don't know who to use the A-bomb against first, Germany or Japan...
Late 19th/early 20th centuries - Gregor Mendel's pioneering work in genetics attracts even more interest than it did OTL. Some prominent scientists and sociologists begin serious speculation about the possibility of improving the human race through selective breeding. Very quietly (we're still in the Victorian era when the TL begins, after all) a program is organized where people with desirable characteristics - exceptional intelligence, strength, physical attractiveness, etc. - are mated to each other.
Circa 1920 - Albert Einstein (or some other prominent physicist of the time - I'm open to suggestions) is doing some work on the nature of space-time, when he is thunderstruck to find that, if his calculations are correct, it is in fact theoretically possible to evade the speed-of-light barrier by "warping" space, thus making it possible to travel interstellar distances in weeks or months instead of plodding along at a fraction of
c. Einstein doesn't release these findings publicly, but discusses them with colleagues he particularly trusts.
1932 - Edith Keeler, a well-known social worker, is killed in a tragic accident when she is hit by an automobile while crossing a street in downtown New York. The police search for two witnesses, a man who was known to be keeping company with Keeler and his vaguely Oriental-looking friend (who is noticeable because of the knit cap he always wears), but they seem to have literally vanished from the face of the earth.
1941 - Hitler decides to screen the shattered Soviet forces in the Ukraine with Army Group South instead of wheeling south to mop them up, choosing to go straight for Moscow. Guderian's Panzers smash into the city just as the snows begin to fall. In the chaos of evacuation to Kubyshev, Stalin is overthrown by Beria, who is himself then deposed and executed by a junta led by Zhukov and Molotov. Zhukov stabilizes the line east of Moscow with troops brought west from Siberia. On Dec. 7th, Japan conducts its Pearl Harbor raid and the US enters the war.
1942 - Hitler vacillates for weeks about whether to order von Paulus to stand fast in Stalingrad or evacuate. Finally, he decides on evacuation and orders von Manstein to cover the Sixth Army as it fights its way out. The fighting is brutal, but the Sixth Army comes out still a coherent fighting unit. In the meantime, German research on rocketry has been proceeding apace. Von Braun has, in what spare time he has, begun noodling around with the problem of manned spaceflight. He also pushes study on the A-9/A-10 multistage rocket in addition to the V-1 and V-2. On another front, other German scientists are conducting researches into eugenics under Nazi Party auspices. The ostensible purpose is to breed super-Aryans, but some of the eugenicists have kept very clandestine contacts with colleagues in the West and are passing their findings to them.
1943-46: Due to various technical problems, the Manhattan Project experiences delays which make a finished nuclear weapon unavailable until late 1945.
The Soviets, under the Zhukov/Molotov junta, grind westward, pushing the fascists back inch by inch. The Germans choose to go on the defensive instead of attacking at the Kursk salient, and the resulting fight, while a narrow tactical victory for the Soviets, leaves their best armored units shattered, and the Germans are able to withdraw into more compact defensive lines. No salient forms in the Army Group Center area that the Soviets can exploit. The Red Army, all the same, keeps grinding relentlessly west, with copious Lend-Lease aid from the US.
The Pacific War, North Africa and Italy proceed much as OTL. Preparations for Overlord, the invasion of Normandy, begin in earnest.
On June 6, 1944, Operation Overlord begins as US, British and Canadian forces storm ashore at five beaches in Normandy. However, the Germans have a new weapon ready. Several dozen V-1 flying bombs are launched against the narrow beachheads. While not very accurate at all, they cause a lot of consternation in the crowded quarters. Coupled with extremely stiff German resistance, discretion is decided the better part of valor and Omaha Beach is evacuated. However, Utah Beach and the Anglo-Canadian beachheads (Gold, Juno and Sword) hold on, and the Allies have their Normandy lodgement, though just by the bare skin of their teeth.
As 1944 proceeds, the Western Allies and Soviets claw forward against the Third Reich from west and east. The Germans conduct tenacious and tactically brilliant defensive operations on both fronts, slowing the advance down to a crawl almost at World War I speeds. V-1 and V-2 missiles rain down on military and civilian targets on both fronts. The Americans put everything they can into the Manhattan Project, which is now seen as being their best chance to break the Third Reich, but technical obstacles are proving stubbornly difficult to overcome.
In late 1945, as the front in Europe comes to a stalemate at the frontiers of Germany, the US finally launches Operation Olympic, the invasion of Kyushu.
Near the end of 1945, though, everything finally comes together. The Trinity test is carried out successfully at Alamogordo, New Mexico, on December 10, 1945. A fierce debate immediately follows as to whether to use the atomic bomb first against Germany or Japan.
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It may not be nearly as realistic as some other TL's, but this is, after all, Star Trek's backstory we're talking about.

As you can see, WWII in Europe has already diverged significantly from OTL. Among other things, what I'm trying to do here is exhaust the Soviets to the degree that they can't exert the same degree of control over the Eastern European nations that they did OTL. If WWII in Europe lasts longer, the reforming West European governments may also be able to raise more troops to fight the Germans, which will have effects down the line as western Europe becomes an independent power center after the war.
A key date I need: when was Khan born? More to come soon as soon as I think it up!
-Joe-