  KH 15D is a protostellar binary system that shows a peculiar light curve. In
order to model it, a narrow circumbinary precessing disc has been invoked, but
a proper dynamical model has never been developed. In this paper, we
analytically address the issue of whether such a disc can rigidly precess
around KH 15D, and we relate the precessional period to the main parameters of
the system. Then, we simulate the disc's dynamics by using a 1D model developed
in a companion paper, such that the warp propagates into the disc as a bending
wave, which is expected to be the case for protostellar discs. The validity of
such an approach has been confirmed by comparing its results with full 3D SPH
simulations on extended discs. In the present case, we use this 1D code to
model the propagation of the warp in a narrow disc. If the inner truncation
radius of the disc is set by the binary tidal torques at {\sim} 1 AU, we find
that the disc should extend out to 6-10 AU (depending on the models), and is
therefore wider than previously suggested. Our simulations show that such a
disc does reach an almost steady state, and then precesses as a rigid body. The
disc displays a very small warp, with a tilt inclination that increases with
radius in order to keep the disc in equilibrium against the binary torque.
However, for such wider discs, the presence of viscosity leads to a secular
decay of the tilt on a timescale of {\approx} 3000 ({\alpha}/0.05)^(-1) years,
where {\alpha} is the disc viscosity parameter. The presence of a third body
(such as a planet), orbiting at roughly 10 AU might simultaneously explain the
outer truncation of the disc and the maintenance of the tilt for a prolonged
time.
