Both sentences could be correct, but they would mean different things.
Turn your head 15 degrees.
These means that, whatever direction your head happens to be facing, you should turn it 15 degrees more in a certain direction.
Turn your head at 15 degrees.
This assumes that there is a reference point, or a known starting position, that would be considered 0º, and you would turn your head to the position of 15º. Another valid preposition would be to:
Turn your head to 15 degrees.
(I'd be more inclined to use to, but I wouldn't deem at "incorrect".)
As far as saying this to a person, you could refer to any body part that rotates (foot would imply turning the ankle, hand would imply turning the wrist, head would imply turning the neck, etc.)
Also, a number like 15º is a small rotation. I imagine most people would use numbers like 45º or 90º more often, but I suppose there are situations that would call for smaller increments, like a therapist trying to help a recuperating patient regain a range of motion, or a photographer trying to help a model pose. The implication would be approximately 15º.
Lastly, your sentences may be correct, but they are quite vague. Unless two people have been working together for quite some time and the context is obvious, I think you'd be likely to encounter a more detailed sentence explaining which direction the movement should go. So instead of:
Turn your feet 15 degrees.
we'd be more likely to hear something like:
Turn your feet 15 degrees more to the left.
Also, instead of:
Turn your feet at 15 degrees.
we might be more likely to hear something like:
Turn your feet to a position of about 11 o'clock.
A: The starting position of the feet.
B: The position of the feet after saying, "Turn your feet 15 degrees."
C: The position of the feet after saying, "Turn your feet at (or to) 15 degrees."