Nothing in the rules as they currently exist bars silvering an adamantine weapon, or any weapon of any material.
However, given that the XGtE was published after the PHB, there is a possibility that the game designers did not really think carefully about this combination. The silvering process is not clearly detailed, meaning that the way it is performed is left to the gamers. The actual answer might depend on the metaphysics of your own gaming world (meaning your DM).
If your fantasy world has similar chemistry/physics as our daily world (which is not an unreasonable assumption for ordinary objects: swords are still metal for example, or gold is malleable so you can make coins, etc), you need to make a decision about what sort of material adamantine is. If the "silvering process" on metals is similar to silver plating, there are some good news: silver can be deposited on nearly any metal. Moreover, while silver plating technology was developed in the 18th century in the real world, so-called Sheffield plating is simple enough that you could imagine it was invented earlier in your fantasy world. If you decide that adamantine does act as a good substrate (like copper), Sheffield plating becomes possible. On the other hand, if you decide it is not a good substrate, you need the invention of electroplating (or the fantasy equivalent of that) to silverplate adamantine.