The Thargoid 'bases' are 3D meshes which have been created and pasted at specific locations. The procedural terrain generation used by the game is heightmap based and so in its current iteration cannot do overhangs, caves or anything with a 'roof' (i.e where there would need to be two or more separate heights at the location of a single pixel's X,Y coordinates. In view of that I'd say 'not any time soon.
Yeah, I know that planets are just heightmaps in Elite so horizontal deformation isn't possible. However, just like the alien bases are placed 3D meshes, I had hoped that planet caves could be implemented similarly into Elite. Create a couple dozen cave "segments" as 3D meshes (a bunch of corners, openings, and straight sections), segments which can be used like building blocks to create varied and randomized caves, and then place those assets via the Stellar Forge just like geysers and such are today and colorize them per the planet they are on.
When David talked about caves in Elite that's the potential implementation that came to my mind, as it seemed like the most likely way Frontier could do it. Now though, having seen the Thargoid bases, I'm left wondering if these are the "caves" David meant, or if there is more coming yet regarding caves in Elite Dangerous.