it will probably never be clearly know who the first successful lunar explorers were, nor for which side they served. In a more peaceful world, perhaps, the moon landing would have been a public display of national will, or technology- essentially a piece of dramatic propaganda to humiliate the opposition. In reality, the moon landings were military missions to gain and map the new high ground, and as such, as secret as possible.
It is known that across 1965 and 66 both east and west used nuclear propelled heavy lifters to orbit and then land materials on the moon to build the national bases, redoubts and missile silos. What is also clear is that crews were already present on the moon, and had been for some time.
While theoretically at peace, or at least ceasefire, both sides actively attempted to sabotage or impede the other sides construction once on the moon. While only partly successful (both sides successfully installed military bases by 1967), this set the tone for the constant low level skirmishing on the Lunar surface.
While only a few missiles were installed by either side, they were uniformly armed with the massive new fusion boosted warheads that had been developed by the end of the orbit wars, making them potent final strike weapons. Initially intended as deterrence weapons, their potential as first strike weapons had the actual effect of drastically destabilizing the shaky ceasefire between East and West.
By 1968, tensions were rising once again, and the lunar skirmishing was spreading once again to earth's orbitals. Given that both sides had deployed ICBMs earthside with fusion warheads it seemed fated that the next outbreak of war would be conclusive, if not survivable.
However, in October of 1968, everything changed.