Fifth generation wireless networks are expected to provide advanced capabilities and create new markets. Among the emerging markets, Internet of Things (IoT) use cases are standing out with the proliferation of a wide range of sensors that can be configured to continuously monitor and transmit data for intelligent processing and decision making. Devices in such scenarios are normally extremely energy-constrained and often exist in large numbers and can be located in hard-to-reach areas; the fact that necessitates the design and implementation of effective energy-aware data collection mechanisms. To this end, we propose the utilization of unmanned aerial vehicles (UAVs) to collect data in dense wireless sensor networks using projection-based compressive data gathering (CDG) as a novel solution methodology. CDG is utilized to aggregate data en-route from a large set of sensor nodes to selected projection nodes acting as cluster heads (CHs) in order to reduce the number of needed transmissions leading to notable energy savings and extended network lifetime. The UAV transfers the gathered data from the CHs to a remote sink node, e.g., a 5G cellular base station, which avoids the need for long range transmissions or multihop communications among the sensors. Our problem definition aims at clustering the sensors, constructing an optimized forwarding tree per cluster, and gathering the data from selected CH nodes based on projection-based CDG with minimized UAV trajectory distance. We formulate a joint optimization problem and divide it into four complementary subproblems to generate close-to-optimal results with lower complexity. Moreover, we propose a set of effective algorithms to generate solutions for relatively large-scale network scenarios. We demonstrate the superiority of the proposed approach and the designed algorithms via detailed performance results with analysis, comparisons, and insights.