||We study the numerical approximation of time-harmonic, electromagnetic fields inside a lossy cavity of uncertain geometry. Key assumptions are a possibly high-dimensional parametrization of the uncertain geometry along with a suitable transformation to a fixed, nominal domain. This uncertainty parametrization results in families of countably parametric, Maxwell-like cavity problems that are posed in a single domain, with inhomogeneous coefficients that possess finite, possibly low spatial regularity, but exhibit holomorphic parametric dependence in the differential operator. Our computational scheme is composed of a sparse grid interpolation in the high-dimensional parameter domain and an Hcurl -conforming edge element discretization of the parametric problem in the nominal domain. As a stepping-stone in the analysis, we derive a novel Strang-type lemma for Maxwell-like problems in the nominal domain, which is of independent interest. Moreover, we accommodate arbitrary small Sobolev regularity of the electric field and also cover uncertain isotropic constitutive or material laws. The shape holomorphy and edge-element consistency error analysis for the nominal problem are shown to imply convergence rates for multilevel Monte Carlo and for quasi-Monte Carlo integration, as well as sparse grid approximations, in uncertainty quantification for computational electromagnetics. They also imply expression rate estimates for deep ReLU networks of shape-to-solution maps in this setting. Finally, our computational experiments confirm the presented theoretical results.