Mewtwo cannot be found anywhere else other than the Cerulean Cave. Given that you have access to the cave I would assume you completed the game, which means that Mewtwo should be down there somewhere.
If not, perhaps you caught or battled Mewtwo at another point in time, and forgot? Or perhaps you traded said Mewtwo away without remembering?
Just to make sure, try rebattling the Elite Four. If you did defeat Mewtwo before, it should respawn in the proper place after defeating the Elite Four again. However if you have already caught it, it won't reappear. Try checking your pokedex and see if you've already seen or captured a Mewtwo.
This is the only logical answer that I can give. If you wish for me to remove it, I will, or convert it into a comment.