This paper is devoted to the numerical solution of the instationary Maxwell equations in waveguides with metallic conical inclusions on its internal boundary. These conical protuberances are geometrical singularities that generate in their neighborhood, strong electromagnetic fields. Using some recent theoretical and practical results on curl-free singular fields, we have built a method which allows to compute the instationary electromagnetic field. It is based on a splitting of the spaces of solutions into a regular part and a singular one. The singular part is computed with the help of a multiscale representation, written in the vicinity of the geometrical singularities. As an illustration, numerical results in a rectangular waveguide are shown.