I don't think that this is particularly new ... Firefox has intentionally not exposed itself as an "intent" ( ... I believe that is the term ... ) for opening local html files for a while ... I'm not sure if it is for security reasons or difficulty dealing with Googles content:// URIs. I think they may have recently gotten rid of other UI features to open local files which is kind of annoying. Chrome should work though ... just not the stripped down Chrome viewer that sometimes pops up to preview stuff ... you will have to choose "Open with" from there and choose Chrome.
However, Firefox can open file:/// URLs. Typing "file:///sdcard/" (notice the 3 x "/") into the Firefox location bar should get you a directory listing of your flash memory. Then you can browse to the location of your tiddlywiki and open it and bookmark it or "Add to Home screen" and treat it like an app. The awkward bit is saving changes since it will just download to a non-conflicting filename ... adding (1), (2), etc. and you will have to go through an awkward rename loop.
The "sdcard" part of the path might be different on your device ... it could be "file:///storage/emulated/0/" or something like that. The full path is discoverable using some third party file manager utilities.
/Mike