Ultra-wideband (UWB) radar imaging can provide high-resolution images of obscured objects using radio-frequency signals. Due to its vast applications, UWB radar imaging received considerable attention in the past decade. Compressive sensing (CS) has been used as a viable solution for the larger data required by radar imaging. The advances of CS-based UWB-radar imaging is burdened by the complexity of the reconstruction algorithms and their weak noise immunity. Exploiting the structure of the basis-matrix, a low-complexity Bayesian-based estimation algorithm is proposed. The algorithm takes advantage of the radar-return statistic to find an approximate minimum mean-square error estimate of the radar image. The low complexity is achieved by utilising the block-matrix-inversion formula to execute the algorithm in an order-recursive manner. Further simplification is achieved by using exponential-sum formula to find the correlation between the columns of the basis-matrix. The proposed algorithm is evaluated over experimental and simulated data. The results show faster processing time compared to other known algorithms, with comparable reconstruction quality.