For version older than 4.4.9:
These files ("gallery-template.php" and "gallery-single-template.php") should be located in your theme folder in order to make it work with your theme. Theme folder is totally replaced with a new folder/files after updates. That is why you get this error after updating your theme.
Solution:
When updating your theme, copy these template files from Gallery plugin folder
yourwordpresstheme.com/wp-content/plugins/gallery-plugin/template/
to your theme folder
yourwordpresstheme.com/wp-content/themes/yourwordpresstheme/
If you are updating the active theme, you can deactivate and activate Gallery plugin in order to copy these files automatically.
This article related to Galley by BestWebSoft plugin
Additional resources: Gallery - User Guide
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