Now all the saving operations work in the following way:
1) Make a shallow copy of the image
2) Create a separate document with it
3) Save it in a separate thread in background

The functionality basically works, except the following parts:

1) Cloning the image "forgets" some image stuff. I don't have a list of what
   is forgotten yet (layer styles?).
2) Closing the image while Krita is saving will most probably cause a crash.
3) D&D of layers is not tested, though it should work.
4) There was some option to provide "real name" for json export plugin,
   it is not supported anymore.