This paper presents an approach to mutual collision avoidance based on Nonlinear Model Predictive Control (NMPC) with time-dependent Reciprocal Velocity Constraints (RVCs). Unlike most existing methods, the proposed approach relies solely on observable information about other robots, eliminating the need for excessive communication. The computationally efficient algorithm for computing RVCs, together with the direct integration of these constraints into the NMPC problem formulation at the controller level, allows the whole pipeline to run at 100 Hz. This high processing rate, combined with modeled nonlinear dynamics of the controlled Uncrewed Aerial Vehicles (UAVs), is a key feature that facilitates the use of the proposed approach for agile UAV flight. The proposed approach was evaluated through extensive simulations emulating real-world conditions in scenarios involving up to 10 UAVs and velocities of up to 25 m/s, and in real-world experiments with accelerations up to 30 m/s$^2$. Comparison with the state of the art shows 31% improvement in terms of flight time reduction in challenging scenarios, while maintaining a collision-free navigation in all trials.

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