An effective maintenance strategy to cut back maintenance costs and production loss with assured product quality has always been a major concern for industries. The Industry 4.0 era has built a wide acceptance for the predictive maintenance techniques in the remaining useful life (RUL) estimation of critical industrial systems. In this paper, long short-term memory (LSTM) and bidirectional-LSTM (bi-LSTM) deep neural architecture-based predictive algorithms are proposed for the RUL estimation of the lathe spindle unit. The deep learning algorithm is embedded within a Bayesian optimization algorithm for the self-optimization of its network structure and hyperparameters. The proposed deep learning algorithm is trained using lathe spindle health degradation data collected from an experimental accelerated run-to-failure test rig to evolve an RUL prediction model. The vibration signals representing lathe spindle health degradation from the health to faulty state are analyzed to extract time, frequency, and time-frequency domain features, which are then subjected to a neighborhood component analysis (NCA) based feature selection criteria. Finally, the selected relevant features are used to train the optimized LSTM/bi-LSTM network for RUL estimation. A comparison of the prediction results for Bayesian optimized LSTM/bi-LSTM network architectures and other prominent data-driven approaches are performed. The Bayesian optimized LSTM + bi-LSTM deep network architecture is observed to have the highest prediction accuracy for lathe spindle RUL estimation.