I don't have that problem on my Surface running Windows 10.
What happens if you log on to that same computer with another user? What happens if that user logs on using another computer?
If user logs in to another computer with she is able to get the publish button and publish. I have not seen how it looks like on the surface, can it look different and due to that the user cannot see the button or understand it is there?
are you try another computer and repeat it ?
Are there known editoral issues with Microsoft Surface win10? A customer that has a Surface win10 using Chrome do not get the publish button after editing.