Before the Cold War began, aliens attempted their first invasion. This would have been around the end of WWII, over Germany. The aliens actually struck Berlin and shredded most of the Nazi party. The Battle of Berlin was actually the first battle against the aliens. Hitler's body was never found because the allies didn't kill him - he had already been taken out by the alien invaders.
The allies (notably the Russians and the Americans) swept up the alien forces, threw them back into their wormhole, and used this nifty new bomb the Americans had, to destroy the wormhole. The atom bomb had been in planning since the first signs of unfriendly alien inspection in the 1940s and was, in fact, not ready to be used as a bomb.
To destroy the wormhole, a pilot crew had to fly a transport plane full of plutonium and uranium into the wormhole and start the chain-reaction by hand. The aliens, being smarter than your average muskrat, had opened a wormhole over Japan as well, though this hole had been closed after the alien forces were dropped. Hiroshima and Nagasaki were hotbeds of alien forces, so the US scrambled to put together more atom bombs. They succeeded and obliterated both cities which, along with the clean-up in Germany, destroyed all evidence and killed all witnesses of the alien invasion.
Most of Europe was in tatters, the rest of the world didn't know about the alien attack and the US and the USSR were the only countries equipped to handle the assault and clean-up. Because tensions between the two countries had been high, the governments could not openly declare an alliance without explaining to their people that aliens were invading.
After two Great Wars, neither side was eager to inform the people of an alien invasion, so both sides kept the deal on the hush-hush. Because the USSR was closer to the epicenter of the invasion, it naturally reached out to protect the shattered Europe from the predicted second attack.
The media in the US spun this as an act of Communist aggression, which the US government embraced and encouraged, creating public support for a build-up of the only weapons known to be effective against the aliens: nukes.
“It must be the policy of the United States, to support free peoples who are resisting attempted subjugation…by outside pressures.” (Harry Truman) - Not exactly a lie, but let's look at the subtext. The USSR took its cue from the US and began pushing anti-US propaganda, also building up an arsenal.
Meanwhile, both sides were hunting down the remnants of the alien forces, most of which had fled Germany and Japan by this time. The governments identified these fugitives as Nazi scientists, thus giving the people an explanation for why soldiers were tearing around South America, shooting people no one had ever heard of. Most of the following "proxy wars" between the USSR and the US were actually missions to roust pockets of the alien army.
The Bay of Pigs was actually a covert operation aimed at an alarmingly caffeinated group of aliens attacking the coffee fields and was a total success. Unfortunately, the Cubans had no idea what was going on, so the surviving forces were captured, leading to a Russian-attempted rescue mission.
Due to a few crossed wires, the US publicized the mission's cover (placing nukes in Cuba). This almost destroyed the entire cover-up as Pres. Kennedy was forced to issue an ultimatum and the Russians were forced to turn back.
What people don't realize is that nukes in Cuba was actually the US's idea to protect its southern flank in case of another alien invasion. We also wanted to place weapons in Canada, but they refused. In short, no missiles went to Cuba, however, the 'missile crisis' created enough of a diversion to spring the Bay of Pigs survivors, so no second attempt was made.
As years passed without another attempted invasion, both the US and the USSR began to reduce their nuclear stockpiles to reduce the risk of accidental discharge. Eventually, the USSR fell apart, ending the Cold War and the US/USSR alien task force alliance. At that point, the US began to drastically cut back its weapons because the new Russian government knew nothing of the alien war and, the Cold War guise having been so effective, we feared the new government might actually take it seriously.
The anti-alien preparations are still in place, and every so often, a new pocket of aliens rears its head, but the US mostly flies solo. The War on Terror is actually a cover to allow the US to take out both individuals and groups of supposed terrorists (actually the residual aliens and perhaps their human supporters). Maybe Russia is getting back into the battle? Going into Syria with us?
In short, massive cover-up began as a fortuitous accident and both war-weary sides just went with it, using the other country as an excuse for their anti-alien operations.
If you're not totally set on the idea that the USSR and the USA actually did repel the aliens, it would be pretty easy to back up a story in which the aliens were able to infiltrate both governments and gain control.
Perhaps the reason the Cold War never turned into a Real War was that the aliens had control of both sides. The build-up of nuclear weapons could be a plan on the part of the aliens; use the human technology to destroy the human defenses.
The whole mutually assured destruction thing might have thrown a wrench in their plans because the other humans in the war councils wouldn't let the Cold War turn into a real war. The several times that it almost did could be attempts by the aliens to reclaim the situation.
After the Cold War ended, the two sides, working together, launched a series of pointless, high-casualty wars intended to weaken and destroy the humans' two largest super-powers before launching their own invasion.
