Delayed-acceptance Metropolis-Hastings (DA-MH) and delayed-acceptance pseudo-marginal Metropolis-Hastings (DAPsMMH) algorithms can be applied when it is computationally expensive to calculate the true posterior or an unbiased stochastic approximation thereof, but a computationally cheap deterministic approximation is available. An initial accept-reject stage uses the cheap approximation for computing the Metropolis-Hastings ratio; proposals which are accepted at this stage are then subjected to a further accept-reject step which corrects for the error in the approximation. Since the expensive posterior, or the approximation thereof, is only evaluated for proposals which are accepted at the first stage, the cost of the algorithm is reduced. We focus on the random walk Metropolis (RWM) and consider the DAPsMRWM, of which the DARWM is a special case. We provide a framework for incorporating relatively general deterministic approximations into the theoretical analysis of high-dimensional targets. Then, justified by a limiting diffusion argument, we develop theoretical expressions for limiting efficiency and acceptance rates in high dimension. The results provide insight into the effect of the accuracy of the deterministic approximation, the scale of the RWM jump and the nature of the stochastic approximation on the efficiency of the delayed acceptance algorithm. The predicted properties are verified against simulation studies, all of which are strictly outside of the domain of validity of our limit results. The theory also informs a practical strategy for algorithm tuning.