DT stores the edits in an .XMP sidecar file, so if you look at a directory in windows with your image lets say it
is named 123.raw, you will find another file named 123.raw.xmp, It is this sidecar file that contains the edits
In fact it is possible (although there may be some limitations) in sharing the image and edits between Windows and Linux simultaneously if you wanted to use them both as Windows will have a library database, as will ubuntal and they will share the xmp file when loaded, although of course both editing the same image at the same time would be foolish. However I do not think you can share just one library database between systems, but I could be wrong.
Now to your question. If you import the folder that contained the image and corresponding xmp file that was created in windows into your ubuntu DT, it should detect the image edits in the xmp file and it should be the same as the windows version.
If this did not work, go to the DT preferences, core options, XMP section and check on the option to "look for updated XMP files on startup" then close and re-start DT