Q: Access TextEdit documents on iCloud from iPad or iPhone

Use Pages.

You can only use TextEdit documents uploaded to iCloud with another Mac with Mountain Lion, but you can use Dropbox to upload the TextEdit documents to the cloud and use them on your iPhone and iPad

Yes, definitely use any other file sharing tool. Apple's iCloud file sharing is pretty much brain-dead.

I just experimented once with a simple plain-text file (no formatting - just text) using Textedit on my Mac, saved it to iCloud. Now, the file does get saved all right: I can see it is there on my iPhone and iPad. But that's where it stops. When I open it on the iPhone, the text file gets opened - guess what ? Using Numbers (Apple's idea of a spreadsheet) - with one cell per line of the original text. Had it been opened using Pages (Apple's idea of a document editor), that would have made some sort of sense. But Numbers ? Come on !

So I just copied the file to my Dropbox. That works much better: I can see the text as it should - plain text.

I can only shake my head and wonder what goes on in the minds of Apple's developers ...

If those TextEdit documents are saved as Rich Text Format (RTF) documents on iCloud, then you will not be reading them on iOS with any Apple application.

They are not rich text. Just plain text. 

But then again, Dropbox and Google drive are both happy to read those rtf files too. Surprisingly, iCloud is also happy to open RTF files! It just barfs on plain text files. Go figure.

BTW this is just for sharing some simple text for viewing. Not for editing on iOS.