I suspect the break angle over the nut will be insufficient and will mean the strings pop out of the slots.
I would:
1. Not panic
2. As suggested, drop a sliver of toothpick in with a bit of glue, leave to dry, cut off So level with the headstock face, make a minor indent in to get the screw in and then re-screw the tree in.
3. If you have one, I'd use a slightly larger screw (tiny bit longer and tiny bit bigger diameter)