I guess you are referring to yEd Live here.
Indeed, the autosave feature of yEd Live periodically saves the graph to the browser's storage. This is also indicated by the 'Unsaved Changes' text that appears below the document title in the header of yEd Live.
If the autosave failed for whatever reason, the periodical auto-save mechanism will be disabled. Depending on why the autosave failed, it may either be that the stored graph is still the most recent working file or it may also happenn that the file in the browser's storage is corrupted and won't work anymore.
There is no UI to directly download the auto-saved graph. When you start yEd Live and an auto-saved graph is found, the "Continue Last Session" button in the welcome dialog will be enabled.
Is this button is enabled in your case? If so, what happens if you continue the last session?