While we wait for the publishing problems at Office Online to be fixed, try this workaround – it finally worked for me after several tries.
Go to office.com and sign in with the same Live ID you use to publish your calendar and select "Save my e-mail address and password". Restart Outlook and the browser. It should now work for either publishing or updating a subscribed calendar, at least until you sign out of Office Online and/or LiveID.
I’m not sure what the secret to making it work is – I trusted Microsoft.com for both web and cookies, set cookie handling to allow all cookies, restarted both Outlook and IE, and deleted saved passwords in Control Panel, User accounts.