We present a method to compute the integrands of one-loop Einstein-Yang-Mills amplitudes for any number of external gauge and gravity multiplets. Our construction relies on the double-copy structure of Einstein-Yang-Mills as (super-)Yang-Mills with the so-called YM+$\phi^3$ theory -- pure Yang-Mills coupled to bi-adjoint scalars -- which we implement via one-loop Cachazo-He-Yuan formulae. The YM+$\phi^3$ building blocks are obtained from forward limits of tree-level input in external gluons and scalars, and we give the composition rules for any number of traces and orders in the couplings $g$ and $\kappa$. One the one hand, we spell out supersymmetry- and dimension-agnostic relations that reduce loop integrands of Einstein-Yang-Mills to those of pure gauge theories. On the other hand, we present four-point results for maximal and half-maximal supersymmetry where all supersymmetry cancellations are exposed. In the half-maximal case, we determine six-dimensional anomalies due to chiral hypermultiplets in the loop.