We study positioning of high-speed trains in 5G new radio (NR) networks by utilizing specific NR synchronization signals. The studies are based on simulations with 3GPP-specified radio channel models including path loss, shadowing and fast fading effects. The considered positioning approach exploits measurement of Time-Of-Arrival (TOA) and Angle-Of-Departure (AOD), which are estimated from beamformed NR synchronization signals. Based on the given measurements and the assumed train movement model, the train position is tracked by using an Extended Kalman Filter (EKF), which is able to handle the non-linear relationship between the TOA and AOD measurements, and the estimated train position parameters. It is shown that in the considered scenario the TOA measurements are able to achieve better accuracy compared to the AOD measurements. However, as shown by the results, the best tracking performance is achieved, when both of the measurements are considered. In this case, a very high, sub-meter, tracking accuracy can be achieved for most (>75%) of the tracking time, thus achieving the positioning accuracy requirements envisioned for the 5G NR. The pursued high-accuracy and high-availability positioning technology is considered to be in a key role in several envisioned HST use cases, such as mission-critical autonomous train systems.