An accurate uniform asymptotic technique to calculate the radiated field from a magnetic current moment situated on the surface of a circular cylinder coated with a thin dielectric layer has been developed. Using the Watson transform, a slowly convergent radially propagating series representation for the fields is converted into a rapidly convergent circumferentially propagating series representation. An asymptotic shadow region solution is then obtained by employing the method of steepest descent. By modifying the shadow region solution in an appropriate manner, a uniform lit region solution is obtained that reduces to the geometrical optics solution in the deep-lit region. The asymptotic solution is valid for large cylinders away from the paraxial region and can be used to calculate the radiation pattern and the scattering from waveguide-fed apertures. Numerical results are compared to an eigenfunction solution, and a very good accuracy is achieved within the expected region of validity.