The tape is smuggled out of the studio they faked it in and it is somehow circulated as proof of the fakes, though the United States broascasts the correct version, this one makes it into the Soviet Union under a re-dubbing and editing to make the Astronauts more Cosmonaut-ish.
It proves to be much cheaper than staging the whole thing before hand and even if the Soviet Union didn't get to the fake moon first, it has the effect of making the United States waste a few million dollars to promote a fake Soviet Moon Landing.
Because of this extra million, the Soviet Union can finance a smaller group in some nations and it tips the balance. By the 1975 , even Afganistan has an Iron-fisted Communist Regime put in place by it's own populace.
The nineteen eighties are a big time for the conspiracy theorists as Neil Armstrong tells about a 'practice run' that they held where he held a soviet flag. He talks about this on national television for the 20th anniversary of the 'Landing'. To screw with the states, the Soviet Union releases the "Special 20th Edition Fake American Moon Landing Blooper Reel" and it hits it big, damaging America's reputation so much that it is laughed out of the UN when proposing an orbital weapons ban extensions to other nations gaining space-capabilities.
Because of this, the Soviet Union rushes it's chance with renewed co-operation between it and China, which didn't sign the treaty to ban weapons in space. Around the last decade's start they have a working set of orbiting rockets.
In 1995 the Korean War restarts and a North Korean mole has hacked the orbital rockets, using them to wipe out concentrations of the troops that would be facing North Korea. The US retaliates with three atomic weapons and China moves to intervene and annexes the entire peninsula, while Taiwan suffers from a coup. Within the next year, it comes into the fold as well. Japan is undergoing a bit of under-the-surface tensions as people wonder if life under communism would be safer than American imperialism for that side of the Pacific, and they point to the Orbiting Rockets as their biggest piece of evidence for kicking the United States out and starting their own actual military again.
In 1997 the British give back Hong Kong, and the Chinese communists got a large boost in trade, and in covert partnership with the USSR, begin to undermine the Republic of India, and Pakistan. The Chinese actually succeed in Pakistan, but India isn't really moving to communism quickly.
By 2000 there are increased votes for communism and acts of violence by radical communists in much of the Middle East. Turkey, instead of looking to Europe, looks to the giant bloc of wealth to the East and signs up due to internal pressure. It will be some time for the Middle East to open up entirely, but the doors are unlocked.
Australia puts down a violent socialist riot brutally and the Russians put pressure on the Middle East to stop the flow of Oil to Australia, as China stops exporting in such a great quantity. Australia relents and the opinions that sparked the riot get large amount of talking about, leading to a further socialist government slowly allying itself with it's closer Chinese neighbor.
I've just begun to scratch the surface, but, you can see how the balance is tipping...
2075: The World Socialist Alliance invades the continental US and shuts off it's Nuclear Silos with a psychic phone call. XP