We propose a novel asynchronous bundle method to solve distributed learning problems. Compared to existing asynchronous methods, our algorithm computes the next iterate based on a more accurate approximation of the objective function and does not require any prior information about the maximal information delay in the system. This makes the proposed method fast and easy to tune. We prove that the algorithm converges in both deterministic and stochastic (mini-batch) settings, and quantify how the convergence times depend on the level of asynchrony. The practical advantages of our method are illustrated through numerical experiments on classification problems of varying complexities and scales.