I've just finished Enforcer and I also couldn't look up or down.
I thought that it was the normal behaviour for the camera... so, in case it's a issue with the GOG release, it also happened to me.

Edit: Just forgot that there is an option to bind a key to “precise aiming” or something like that, that allows you to freely aim, but the angle of the camera doesn't change.

Pretty sure that's a "feature". If you enable "mouselook" in the options, you can aim up/down but the camera remains fixed, so I wouldn't recommend it. The game was made with a fixed camera+aim in mind.

The game's terrible camera is a "feature". The brainiacs behind this game thought it would be preferable to a more standard 3rd or even 1st person view. This wouldn't be a problem except the perspective itself is limiting, and the more you play the more you realize it actually works against you the minute airborne enemies start appearing.

There's a way to change that, though, which surprisingly isn't mentioned anywhere.

Go to <My Drive>:\X-Com Enforcer\System. Track down the User.ini file there and open it. In the file, look for the following line:

Aliases[26]=(Command="Toggle bExtra0",Alias=XFreeLookToggle)

"XFreeLookToggle" is X-Com Enforcer's alias for Unreal/UT's standard Freelook toggling. What this means is that the function exists but if you scour the .ini, it's not mapped anywhere - and it's not available in the game's menus. To change this, simply copy the alias itself (that's "XFreeLookToggle" without the quotes) and associate it with any key you want to toggle. For instance:

Q=XFreeLookToggle

means you can now look freely around when pressing Q during play. Pressing it again defaults to the god-awful perspective.

Keep in mind that the game has a weird issue with the crosshair, in the sense it never centers on screen when playing with Freelook. But it's a much better way to handle some of the later enemy placements.