So.... what was with that gigantic multi-stage moon rocket that was built for show?
Obviously, to fool the international community and any number of private scientists and researchers, you'd have to build or at least dummy up a rocket large and powerful enough to reach the moon. And you'd have to launch it. Since it would be tracked by telescopes, you'd have to keep that rocket going till at least earth orbit, before you could count on people losing sight of it.
And you'd have to probably keep it there, because if you just dropped it out of orbit, there's a good chance it might be detected coming back down. Unless you had multi-stages, some of which were intended to come down.
Also, the Russians, the Chinese, the French, the British, and a great many amateur radio enthusiasts would probably be monitoring the telemetry, and they'd notice an absence. So you would probably need to fake that.
The only reasonable way to fake telemetry of a manned mission to the moon, would be by actually sending a telemetry radio broadcaster to the right spot as a decoy - which would amount to a manned or unmanned mission to the moon. So you'd at least have to send a really complex and sophisticated and fairly large probe there. And it would probably have to land and take off, to preserve the illusion.
Then, you know, you'd have to take the re-entry. So you'd need something the right size and shape for a lunar capsule to descend from orbit. Again, radar and telemetry are your enemies. But assuming you had a package waiting up there, yep.
That's just for the superficial stuff. You'd also have to build your lunar simulation studio in New Mexico. Now, depends on how far into it you wanted to get. There's no way to directly simulate lunar gravity.... unless you had your astronauts all dangling from thin wires as they walked around. But in that case, you'd need a whole cast of trained puppeteers working with your astronauts.
Otherwise, when they try a galileo experiment, some bastard is going to measure the velocity of falling objects, and the jig will be up.
Would the Studio perfectly simulate lunar conditions? Vaccuum? Intense solar radiation. That would be incredibly expensive to design and probably take a few thousand technicians and billions of dollars.
But less might risk exposure.
On the other hand, you could probably half-arse it with a few hundred technicians and hundreds of millions of dollars. Just have an acre inside an enclosed dome, paint it, insert LED for Stars, a sun, landscape the whole thing, and bob's your uncle.
Just fuzz the reception, and hope that no one notices things like air distortion, dust, etc.
I dunno. The problem is that while you could do it - by simply building your giant moon rocket, launching it into orbit, placing a dummy retrieval pod in orbit, and sending a second dummy radio and telemetry transmitter to the moon and back, whilst building a giant studio, equipped with vacuum seals and puppeteers to simulate lunar gravity, you're looking at a conspiracy involving thousands, perhaps tens of thousands of people and all sorts of loose ends.
So what that means is that if they are faked in this Universe... you can expect the secret to hold... a few years at most, and then the cat's out of the bag.