Owing to the deployment flexibility, unmanned aerial vehicles (UAVs), equipped with lightweight base stations (BSs), provide a new paradigm for wireless access from the sky to temporarily complement terrestrial networks, which bring benefits to many applications, such high-speed railway (HSR) communications. In UAV-assisted wireless communications, the advantage of the line of sight (LOS) dominant air-ground channels makes it more attractive to apply directional millimeter wave (mmWave) communications, where beam management is highly related to user location. Nevertheless, due to the high mobility of UAVs and trains, how to accurately and efficiently track beams becomes a critical problem to solve. Based on this observation, in this paper, we first present an UAV-assisted dual-band HSR wireless network architecture, which integrates the frequency bands below 6 GHz (sub-6 GHz) to provide reliable transmissions for UAV controls and HSR safety services, with the mmWave bands to enhance the transmission capacity between UAVs and trains. To solve the beam tracking problem, we analyze the beam angular variations, and then propose an efficient algorithm based on Kalman filtering (KF) with varying update periods and long-short term memory (LSTM) to improve the beam tracking performance. Finally, we conduct extensive simulations to demonstrate that the proposed scheme can achieve more accurate beam tracking and higher spectrum efficiency.