We have the answer now. Not sure yet what was changed by the web hosting, but it was simply that the server was caching pages rather than sending refreshed versions. This stopped it showing the new posts to the user who made them. It also made it send out the last version of a page that another user had seen. Solution was to modify the .htaccess to force all pages to be refreshed. In case anyone need that, here's the addition that worked for us:
Header unset ETag
Header set Cache-Control "max-age=0, no-cache, no-store, must-revalidate"
Header set Pragma "no-cache"
Header set Expires "Wed, 11 Jan 1984 05:00:00 GMT"