From looking at your project, it seems the enemy fires when the timer runs out, but in the Timer-node’s properties it is not started. Setting it to autostart makes it fire after 1 second.
I didn’t watch the video all the way through, so I don’t know if the video forgot to mention it or you overlooked it… but simply you need to start your timer!
You can do that by either starting it in the code (in _ready, I suppose) or set the Autostart property of the timer to true so it’ll automatically start counting.