It looks very borderline as to whether you'll need a string tree or not.
You may be able to get away with it if you use a slightly longer-than-normal nut, but you risk the strings binding in the slots so this isn't satisfactory. If the fingerboard hadn't been attached, you could have either added a zero fret (brings the guiding nut position back a bit, changing the string break angles) or reveneered the headstock (if you had more spalt) after dowelling and redrilling the tuner positions.
Personally (and this is in no way a recommendation) I like the straight string pull of your 1st, 2nd, 5th and 6th strings, and some kind of tree to break the angles of the 3rd and 4th out halfway down the headstock would look cool. In my head anyway. It would probably be impractical really. Realistically, the 1st and 6th should have been 1/4-1/3" closer to the nut, and the 2nd and 5th slightly closer also, or something along those lines.
I really hope that you get this hurdle overcome because this looks like a very very nice build otherwise!