The Soviet Union was in fierce competition with the US and beat America early in the race. I don't recall at any time the Soviet Union claiming the landing was fake.
With most conspiracy theories I've read they all have one common problem. The sheer amount of people it would take to be involved in the theory. To fake the moon landing would involved thousands of people. All the astronauts, people at NASA, movie directors, prop handlers, ect.. it would be impossible to keep a secret.