With robots increasingly operating in human-centric environments, ensuring soft and safe physical interactions, whether with humans, surroundings, or other machines, is essential. While compliant hardware can facilitate such interactions, this work focuses on impedance controllers that allow torque-controlled robots to safely and passively respond to contact while accurately executing tasks. From inverse dynamics to quadratic programming-based controllers, the effectiveness of these methods relies on accurate dynamics models of the robot and the object it manipulates. Any model mismatch results in task failures and unsafe behaviors. Thus, we introduce Rapid Mismatch Estimation (RME), an adaptive, controller-agnostic, probabilistic framework that estimates end-effector dynamics mismatches online, without relying on external force-torque sensors. From the robot's proprioceptive feedback, a Neural Network Model Mismatch Estimator generates a prior for a Variational Inference solver, which rapidly converges to the unknown parameters while quantifying uncertainty. With a real 7-DoF manipulator driven by a state-of-the-art passive impedance controller, RME adapts to sudden changes in mass and center of mass at the end-effector in $\sim400$ ms, in static and dynamic settings. We demonstrate RME in a collaborative scenario where a human attaches an unknown basket to the robot's end-effector and dynamically adds/removes heavy items, showcasing fast and safe adaptation to changing dynamics during physical interaction without any external sensory system.