I have problems updating the game to 1.16. After I run the patch it says "Celtic Kings is not installed".

At first I've installed the game in C:\games\Celtic Kings, after that I thought about installing it to the default folder in Program Files, but it didn't help.

Anyone else having problems with patching the game? Any solutions?

Damn, you are right, it's already 1.16.
I didn't know where to watch what version is the game and the Update button was red, so I supposed it isn't the latest and it wanted me to update it.

Anyway I saw that the version number sits just above the logo and it's 1.16.

To check on version numbers you can try the gog wiki, i think the version numbers are listed there somewhere.