Try:
– switching to the Twenty Twelve theme by renaming your current theme’s folder inside wp-content/themes and adding “-old” to the end of the folder name using FTP or whatever file management application your host provides.
– resetting the plugins folder by FTP or phpMyAdmin.
– re-uploading all files & folders – except the wp-content folder – from a fresh download of WordPress. Make sure that you delete the old copies of files & folder before uploading the new ones.
– running the upgrade manually via wp-admin/upgrade.php
First try to deactivate all plugins before you try to update wordpress.
Petermeadit – thanks but I’ve been locked out of admin screens each time I click from site to Dashboard, themes etc I get white screen with above message.
Other option of ‘knowing’ enough to do a manual upgrade was what I was trying to avoid. I will end up loosing the site. That’s what happened with a php forum I tried to upgrade and lost forever
If you cannot manage this yourself, perhaps you need to consider hiring someone to carry out the work for you? All of the steps I suggested above should be do-able by a site oner with basic skills like FTP etc.