Risk assessment is a key issue when handling the increasing use of unmanned aircraft systems (UASs), especially in integrated operational urban airspace. This study proposes a method to systematically quantify the mid-air collision (MAC) risk for different operation types in urban air mobility (UAM). Specifically, a novel assessment framework is formulated that includes three timelines to track risk evolution for integrated operations (i.e., cooperative, non-cooperative, and manned aircraft) and four safety barriers (i.e., procedural, strategic, tactical, and collision avoidance mitigation) to prevent a conflict ending in MAC. Under the framework, analytical models are established considering trajectory uncertainty, latency time, and detection and avoidance (DAA) system risk ratio to calculate the failure probability of each barrier, and thus an overall MAC probability. Result-oriented, process-oriented and comprehensive Monte Carlo simulations are constructed to validate the proposed models and the MAC assessment timeline, followed by demonstrating four operational scenarios in real-world environment to illustrate the assessment process of the method. Results show that the simulation probability curves closely match the theoretical predictions. The cooperative UAS contributes the highest MAC risk in our designed integrated environment, with strategic mitigation failures accounting for the largest proportion, and thus effective strategic trajectory planning is crucial for maintaining the safety of integrated operations.