After much searching, I’ve found a way to import outlines made in MS Word into Omnioutliner. This is for outlines using the Outline View in Word, not for outlines made using a list style, i.e. for outlines using nested Heading 1, Heading 2, etc.
Go to https://devotter.com/converter which is the Universal Text Document Converter. Select your file. Select OPML as the Convert To format. Click Convert.
It will then output a bunch of text into your browser window. Select all of that text and hit Copy.
(It will ask if you want to copy that to the clipboard, which would be convenient, but that didn’t work for me, hence the manual copy).
Open a blank Text Edit or similar plain text file, and copy the text into it. Save the file.
Now change the extension of that file to .opml
Open the document in Omnioutliner, and it should be what you want. Now choose Save As and save the document as an Omnioutliner document, and you are done.
Revision: After you have the OO document, you may need to universal search and replace to get rid of some formatting tags. For example the things that were italicised in my Word outline were not italicised, but were preceded by < em > and followed by < /em > (without the spaces).
If you are doing more than one document, you should reload the page between them to clear the deck, otherwise they will all end up on the same web page and you might end up selecting more than one at a time, which will give you an error in the final file.
WARNING: I just tried this method with a very long outline, and it ended up being garbled, and I couldn’t get it to work on a second and third try. So check carefully that your results are right!