Activating it online didn't work for me...but what did work was going to the stores.office.com, going to My Account, and then clicking on the license I had previously added online (Microsoft Office Professional Plus 2016).
Then there is a little hyperlink called "View your product key". When I clicked this it was NOT the same product key I had originally entered when confirming my license. However, when I typed this new product key in to the Word app to activate it, it all seemed to work.
I'm not sure why this new product key that is associated with my license is different to the one I originally entered, but it seems to work for activating the offline app, when the original key did not.