In complex urban environments, dynamic obstacles and multipath effects lead to significant link attenuation and pervasive coverage blind spots. Conventional approaches based on large-scale fixed antenna arrays and UAV trajectory optimization struggle to balance energy efficiency, real-time adaptation, and spatial flexibility. The movable antenna (MA) technology has emerged as a promising solution, offering enhanced spatial flexibility and reduced energy consumption to overcome the bottlenecks of urban low-altitude communications. However, MA deployment faces a critical velocity mismatch between UAV mobility and mechanical repositioning latency, undermining real-time link optimization and security assurance. To overcome this, we propose a predictive MA-UAV collaborative control framework. First, optimal antenna positions are derived via secrecy rate maximization. Second, a Transformer-enhanced long short-term memory (LSTM) network predicts future MA positions by capturing spatio-temporal correlations in antenna trajectories. Extensive simulations demonstrate superior prediction accuracy (NMSE reduction exceeds 49\%) and communication reliability versus current popular benchmarks.