Thank you!
Okay, so I have spent hours debugging this, and it turns out, that the ngg_images (or any nextgen gallery shortcode) is not registered during the time of the execution of the ajax request.
I have looked through the NGG plugin API to see if there is any way to force, register those modules, but I could not find the proper way to do so. The images are not showing, because the gallery shortcode is not executed, and the plugin cannot parse the images. Every other shortcode is okay, but that gallery does not work.
I am guessing that there is either some sort of exception within the gallery plugin to not to register it’s shortcodes within ajax context, or there is some sort of a precentence related issue, I honestly can’t tell. If you want, you can ask the gallery author, if there is any possible way of initializing the plugin shortcodes programatically.
Maybe there is a function I could call to initialize the modules, but I was not able to find it within the code unfortunately.