Ah, yes. The age-old question.
You see, the bit about a cubone’s skull and club coming from its deceased mother is actually quite misleading: this only occurs with certain wild cubone, particularly those native to notorious grave sites or extreme environments (which is to say, most wild cubone habitats). In these cases, it’s nearly guaranteed that the mother will die shortly after evolving, mating, and giving birth to a single young, as male marowak are typically solitary, and the environment is often too harsh for a mother to defend on her own. The young, meanwhile, may have a chance to make it to maturity by not only toughening themselves through digging out their first armor and weapon but also by virtue of being small enough to take cover from their worst predators. (Note that I say first armor and weapon. We’ll get to that in a moment.) Even then, cubone are notoriously rare in the wild for exactly this reason.
On the other hand, tame and captive-bred cubone are an entirely different story. Naturally, human homes and breeding centers are far, far safer for a pokémon, and because of this, there’s virtually no reason for a cubone to dig out its own mother’s skull. For this reason, cubone that fail to acquire a helmet and club, if left alone long enough (typically overnight), will simply advance to what most wild cubone will do once they outgrow their “baby skulls,” as it were: calcify their own. That is to say, your cubone essentially made his own as quickly as he could by mixing a special substance he secretes from his head with pieces of his own eggshell, dirt, or other debris until it became a skull-like helmet.
As for his club, that’s a bit … less appealing, I’m afraid. It’s likely a stick or other long object, certainly, but in order to calcify it properly, your cubone very likely had to, well, regurgitate that same substance onto it. Luckily, you can replace these with ordinary bones or even rawhide as he grows so he doesn’t have to do that repeatedly.