You can use javascript/jquery to solve this.
When you insert a gallery in a wordpress posts, the whole gallery is wrapped by a div with id like “gallery-1” but also a class that’s always “gallery”.
Also, every item is surrounded by two other “dl” and “dt”, with class “gallery-item” and “gallery-icon” respectively.
So, you can just use jquery to match every link inside those classes, and add whatever lightbox script you want.
If it’s fancybox, i think something like this should work:
jQuery(".gallery-icon a").fancybox();
You can be more specific, matching css classes .gallery .gallery-item .gallery-icon in that order and then the a (for the link).
For the new Gutenberg galleries, this should work:
jQuery(".wp-block-gallery .blocks-gallery-item a").fancybox();
If you want users can navigate between images as a gallery, then use:
jQuery(".gallery-icon a").fancybox().attr('data-fancybox', 'gallery');
And for the new Gutenberg galleries, use this instead:
jQuery(".wp-block-gallery .blocks-gallery-item a").fancybox().attr('data-fancybox', 'gallery');
If you want more grained control (for multiple galleries on the same page), check this response.
Or use a simple plugin that use the same approach from Viper007Bond, that does clean and nicely, but using colorbox instead.