Non-orthogonal multiple access (NOMA) is widely recognized for enhancing the energy and spectral efficiency through effective radio resource sharing. However, uplink NOMA systems face greater challenges than their downlink counterparts, as their bit error rate (BER) performance is hindered by an inherent error floor due to error propagation caused by imperfect successive interference cancellation (SIC). This paper investigates BER performance improvements enabled by reconfigurable intelligent surfaces (RISs) in multi-user uplink NOMA transmission. Specifically, we propose a novel RIS-assisted uplink NOMA design, where the RIS phase shifts are optimized to enhance the received signal amplitudes while mitigating the phase rotations induced by the channel. To achieve this, we first develop an accurate channel model for the effective user channels, which facilitates our BER analysis. We then introduce a channel alignment scheme for a two-user scenario, enabling efficient SIC-based detection and deriving closed-form BER expressions. We further extend the analysis to a generalized setup with an arbitrary number of users and modulation orders for quadrature amplitude modulation signaling. Using the derived BER expressions, we develop an optimized uplink NOMA power allocation (PA) scheme that minimizes the average BER while satisfying the user transmit power constraints. It will be shown that the proposed NOMA detection scheme, in conjunction with the optimized PA strategy, eliminate SIC error floors at the base station. The theoretical BER expressions are validated using simulations, which confirms the effectiveness of the proposed design in eliminating BER floors.