How exactly does the new Mountain Lion filesystem work?

Don't worry, the filesystem is really the same as in Lion. What you may have misunderstood is how the new iCloud integration works. Every app that supports it (it may even be built-in if you use the system-standard open/save components, I'm unsure) will display an iCloud tab in the Open/Save dialogs, but this is in addition to the standard Finder hierarchy that's been around forever.

This new iCloud storage doesn't allow nested folders, so you only get one level of hierarchy. The idea is to keep things simple à la iOS (it looks like and is conceptually the same as organizing apps on iOS).

For more detail and description, check out the iCloud and You section of John Siracusa's Mountain Lion review at Ars Technica.

The filesystem in Mountain Lion is exactly the same as the one in Lion. It's no different.

What you are referring to, is the document storage facility within apps that use iCloud. If you store your documents within iCloud enabled apps, then you are as you say limited to a depth of 2 when it comes to file structure, so you can store all your files at the top level, or you have have folders, but you cannot have subfolders within those folders.

The whole point is to simplify the storage medium, ensuring that files created within apps are held by those apps only, and not nested in some obscure directory structure whereby as Siracusa points out "direct interaction with the filesystem is where usability goes to die"

See this for more detail.