Sparse regression codes (SPARCs) are a class of codes that encode information through the superposition of columns of a randomised coding matrix. The combination with an outer non-binary low density parity check (NB-LDPC) code was recently shown to improve the finite-length performance of these codes over the unfaded AWGN channel. In this paper, we propose a low-complexity approximate message passing (AMP) decoder that is capable of decoding NB-LDPC encoded SPARCs on a Rayleigh fading channel with multiple receive antennas. Notably, the decoder does not require channel state information (CSI), i.e., it is fully non-coherent, but achieves the same error probability as a decoder with full CSI, even for moderate block lengths. This is achieved by iteratively re-estimating the channel throughout the decoding iterations. In addition, we provide a rigorous asymptotic analysis of both the block error probability and the channel estimation error. Numerical results confirm the precision of the analysis and show that the presented coding scheme performs within 1.5 dB of the outage capacity and is competitive with coded modulation schemes employing standardised LDPC codes for 5G cellular networks and pilot-based channel estimation.