I think you answered your own question.
There is no wood on the cutaway side, thus there is room for the neck to move.
As far as I can tell, all you have to do is pull the neck towards you (if you're sitting at it like in the picture). This should place the neck perfectly in the pocket.
And if your template was placed correctly, the neck will be aligned with the body. If your neck is aligned with the body now, then you have some issues...