Lovecraft died in March 1937, at age 46. He would have been 55 just before the end of World War II, and 67 when Sputnik launched. That's a reasonable lifespan for someone in that era, and I think things would have changed greatly for Lovecraft in those twenty years. Certainly the world did.
In May 1937, there was the Bombing of Guernica. Later on of course, there was the buildup to World War II, the Holocaust, etc. There was industrialized killing of civilians, massive airstrikes that burned down entire cities, and finally the atomic bomb. Then the Soviets acquire their atomic bomb, the United States develops the thermonuclear bomb, and by 1957 Sputnik 1 is launched. The late 1950s also sees the growth of world travel and trade.
By the 1950s, Lovecraft's Cthulhu Mythos would be rather antiquated. Exotic places aren't that exotic with international travel. Eldritch horrors would seem quaint compared to the horrors of World War II and the emerging horrors of thermonuclear weapons. However, exploration of some very remote areas would seriously ramping up, such as space, underwater, and Antarctica. I think there are a few routes that Lovecraft could take in such an environment:
1. Keep writing something similar to the Mythos, but change to underwater and space. After World War II, the world is less exotic. Someone could fly to Alaska, Tibet, or the Pacific is so inclined. It would be expensive in the 1950s, but quite possible.
2. Technology could replace eldritch horrors. It could even follow the same basic formulas. Someone (or perhaps humanity as a whole) feels a need to invent something or know something. They push ahead, even though there are signs it might not be the best idea. Then the technology leads to something terrible happening. Alternatively, people could be the eldritch horrors, and society could misuse the technology to create something terrible.
As an example, perhaps a man of science sets out to develop robotic components to help his wife suffering from an incurable disease. However, he has to race to develop more and more components to avert the fate. Eventually, his wife is almost entirely robotic. Then he realizes he can't replace her brain. All the technology has done is prolong the inevitable, and inflict a horrible fate.
3. Something akin to the Dreamworld phase, but focused more on psychological horror. The noble man of science unleashed the atomic genie, what other horrors await? What's more fearsome than thermonuclear war? What about the experiences of explorers as they venture from the safety of civilization into the depths of the oceans and space?
4. Perhaps he takes a more Lovecraft-lite take to things, and writes from a more existentialist viewpoint instead of a nihilistic one.