I'm guessing the answer to this is that I'm an idiot, but I'll ask anyway. I don't see a health bar anwhere on the screen or a PDA of any kind. Is there supposed to be? With other games like the TR series changing resolution can push that off the screen so I played with that but no dice.

EDIT
Found it!! Far out. Turns out I'm an idiot after all. Happy New Year everyone!

Nobody is an idiot here but the GOG team certainly did some weird stuff in case of Shadow Warrior. By default you should see the full HUD bar on the bottom of the screen, for some reason GOG's version has weird custom presets where the HUD is turned off. I guess you already found the solution yourself but in case anybody else runs into problems:

The solution is adjusting the HUD size by using the + and - keys, either the ones next to backspace or the ones on the numpad.