This paper discusses the effect of nonuniform varying communication delay on distributed consensus algorithms in discrete time. After introducing the delayed mathematical model, we first investigate the ergodicity of the delayed system using the properties of scrambling matrices. Subsequently, the effect of nonuniform varying delay on convergence is examined. It is shown theoretically that nonuniform delay is not detrimental to the convergence rate of the algorithm for directed acyclic graphs. The results are also illustrated with several numerical examples.