I bought Neverwinter Nights: Enhanced Edition Digital Deluxe Edition some time ago but now I noticed that I do not see any DLCs to download. Somebody had this problem?

Depends on what you mean by DLC. All the expansions for the game are included in the install, and so are the extra content like King'sguard and Witches Wake. Smaller modules made to show of the creation tool.
If you mean the Premium Modules like Darkness Over Daggerford you can either buy those on GoG or through the game. If you can't find the modules they are located under "New Game" then "Premium Modules."

Beamdogs has also added a few qurrated fan modules and you can access those through the "Community" tab located after "New Game."

Apparently not. I bought the deluxe with ALL the DLCs. Installed. The premium stuff (like Pirates of the Sword Coast) is NOT installed. I cannot choose "Play" but instead there is the option to "Buy". But I did buy them all. I did check it like you said. They are not listed as installed in the game.

If you installed the game through gog Galaxy you can find the settings button to the right of the "Play" button on the gamepage. Manage Installation, Configure..., and then a tab called DLC. See if you have checkboxes filled otherwise the premium modules won't download.

If you're using the offline installers you need to download a installer for every module, you find those in your games library on the site. "Download Offline Backup Game Installers"
When I double checked mine, the site took a few sec to load the list. So don't panic if you can't see the full list right away.

This is exactly the problem. The Offline Backup Installers ARE NOT there! The list is empty!

Ok, then I guess at this point you should contact the gog support team.