In this paper, an advanced phase-field lattice Boltzmann method based on the multiple-relaxation-time collision model is used to simulate the immiscible single-mode Rayleigh-Taylor instability with a moderate Atwoods number in a long tube, and we systematically analyze the effect of the Reynolds number on the interfacial dynamics and the late-time development stages of interface disturbance. The highest Reynolds number in the current simulation reaches up to 10000. The numerical results show that the Reynolds number significantly affects the development of the instability. For high Reynolds numbers, the instability undergoes a sequence of different growth stages, which include the linear growth, saturated velocity growth, reacceleration, and chaotic mixing stages. In the linear growth stage, the developments of the bubble and spike conform to the classical linear growth theory, and it is shown that the growth rate increases with the Reynolds number. In the second stage, the bubble and spike evolve with the constant velocities, and the numerical prediction for spike velocity is found to be slightly larger than the solution of the potential flow theory proposed by Goncharov [Phys. Rev. Lett. 2002 88134502], which can be attributed to the formation of vortices in the proximity of the spike tip. In addition, it is found that increasing the Reynolds number reduces the bubble saturated velocity, which then is smaller than the solution of the potential model. The nonlinear evolutions of the bubble and spike induce the Kelvin–Helmholtz instability, producing many vortex structures with different scales. Due to the interactions among the vortices, the instability eventually enters into the chaotic mixing stage, where the interfaces undergo the roll-up at multiple layers, sharp deformation, and chaotic breakup, forming a very complicated topology structure. Furthermore, we also measured the bubble and spike accelerations and find that the dimensionless values fluctuates around the constants of 0.045 and 0.233, indicating a mean quadratic growth. And for low Reynolds numbers, the heavy fluid will fall down in the form of the spike, and the interface in the whole process becomes very smooth without the appearances of the roll-up and vortices. The late-time evolutional stages such as the reacceleration and chaotic mixing cannot also be observed.