We study an approximation scheme for a nonlinear filtering problem when the state process X is the solution of a stochastic delay diffusion equation and the observation process is a noisy function of X(s) for s is an element of [t - tau, t], where tau is a constant. The approximating state is the piecewise linear Euler-Maruyama scheme, and the observation process is a noisy function of the approximating state. The rate of convergence of this scheme is computed.