We propose a method for a fast and accurate estimation of the collision probability in a short window following the generation of a debris cloud. A multirevolution Lambert targeting problem is employed to determine all the locations where the collision probability is nonnegligible. Then, the Lambert problem and state transition matrix are used to obtain the image of the target body in the spread velocity space, where the collision probability computation is performed. To perform and control the accuracy of the integration of the collision rate needed to obtain collision probabilities, we apply high-order Taylor expansion and automatic domain splitting techniques. The method is validated using a Monte Carlo simulation and a literature-based test case. Test cases demonstrate the validity of the approach for Keplerian and perturbed dynamics.