This letter proposes a design of a distributed prescribed-time observer for nonlinear systems representable in a block-triangular observable canonical form. Using a weighted average of neighbor estimates exchanged over a strongly connected digraph, each observer estimates the system state despite the limited observability of local sensor measurements. The proposed design guarantees that distributed state estimation errors converge to zero at a user-specified convergence time, irrespective of observers' initial conditions. To achieve this prescribed-time convergence, distributed observers implement time-varying local output injection gains that monotonically increase and approach infinity at the prescribed time. The theoretical convergence is rigorously proven and validated through numerical simulations, where some implementation issues due to increasing gains have also been clarified.