This paper proposes a real-time solution method for a variant of the interval load flow problem of seeking extremal bus voltages and line currents for radial power distribution systems described by the nonlinear DistFlow equations subject to uncertainties in bus power injections. We show that a single deterministic load flow run can compute all minimum bus voltages. In addition, we develop a low-complexity computational procedure to estimate all maximum bus voltages and maximum line currents in only one algorithmic run. This leads to significant speedup over standard methods requiring a loop to solve at least one optimization problem for each bus and line. Benchmark case studies demonstrate the accuracy and efficiency of and the need for the proposed algorithm for real-time security analysis.