Try downloading WordPress again, access your server via SFTP or FTP, or a file manager in your hosting account’s control panel (consult your hosting provider’s documentation for specifics on these), and delete then replace your copies of everything on the server except the wp-config.php file and the /wp-content/ directory with fresh copies from the download. This will effectively replace all of your core files without damaging your content and settings.
Some uploaders tend to be unreliable when overwriting files, so don’t forget to delete the original files before replacing them.
Thank you very much for your answer!
Hm…I would be a little nervous to start with that – never knowing, what I could do wrong.
However, in the meanwhile I found out, that it is the theme giving me issues after the update.
So I am thinking about looking for a plugin to roll back the update, until the theme has hopefully fixed it.