It is commonly expected that future fifth generation (5G) networks will be deployed with a high spatial density of access nodes (ANs) in order to meet the envisioned capacity requirements of the upcoming wireless networks. Densification is beneficial not only for communications but it also creates a convenient infrastructure for highly accurate user node (UN) positioning. Despite the fact that positioning will play an important role in future networks, thus enabling a huge amount of location-based applications and services, this great opportunity has not been widely explored in the existing literature. Therefore, this paper proposes an unscented Kalman filter (UKF)-based method for estimating directions of arrival (DoAs) and times of arrival (ToA) at ANs as well as performing joint 3D positioning and network synchronization in a network-centric manner. In addition to the proposed UKF-based solution, the existing 2D extended Kalman filter (EKF)-based solution is extended to cover also realistic 3D positioning scenarios. Building on the premises of 5G ultra-dense networks (UDNs), the performance of both methods is evaluated and analysed in terms of DoA and ToA estimation as well as positioning and clock offset estimation accuracy, using the METIS map-based ray-tracing channel model and 3D trajectories for vehicles and unmanned aerial vehicles (UAVs) through the Madrid grid. Based on the comprehensive numerical evaluations, both proposed methods can provide the envisioned one meter 3D positioning accuracy even in the case of unsynchronized 5G network while simultaneously tracking the clock offsets of network elements with a nanosecond-scale accuracy.