The Antarctic Treaty doesn't establish ownership or sovereignty
Article IV of the Antarctic Treaty makes clear that the Treaty is not a renunciation by any contracting party of any basis of claim to territorial sovereignty in Antarctica. And "[n]o acts or activities taking place while the present Treaty is in force shall constitute a basis for asserting, supporting or denying a claim to territorial sovereignty in Antarctica or create any rights of sovereignty in Antarctica."
So, while the Antarctic Treaty establishes an arrangement for cooperative shared use of the areas, it is not about sovereignty or ownership.
There are (were) however territorial claims to various areas of Antarctica. These have not been resolved at international law.
Who owns the centre of the earth?
A traditional common-law property rule is that ownership extends to the center of the earth (see generally, John G. Sprankling, "Owning the Center of the Earth"). Sprankling notes that this was never taken literally until after it was introduced by Blackstone and argues that its time is near an end. But in any case, I think the correct starting point is that the owner of the surface owns what's underneath, at least to an extent.
Who owns the alien bases, and what would that mean?
Applying Earth law, ownership of the alien bases would be disputed among those nations who are willing to assert territorial claims to the surface under which the alien bases are found. However, if these bases are cultural heritage property (which their age suggests they might be), the 1970 UNESCO Convention on the Means of Prohibiting and Preventing the Illicit Import, Export and Transfer of Ownership of Cultural Property could prevent the import, export, or transfer of any of the material from the bases. The "owning" country (if that is ever resolved) might have obligations under international law to preserve the bases. Further, if the dispute over the property results in armed conflict, the competing states could have obligations under the 1954 Convention for the Protection of Cultural Property in the Event of Armed Conflict to avoid damaging the cultural property.
Is Earth law the applicable law?
If we take this hypothetical at face value, it implies we are not alone and in relationship with other extra-terrestrial culture(s). If members of those cultures were to reach out, that should at least bring us to consider that our law is not supreme. Our law would need to coexist with other law; other law that potentially has very different conceptions of ownership and rights.