This paper proposes a real-time solution method for a variant of the interval load flow problem. In this problem, we seek 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. Also, we develop a low-complexity procedure to estimate all maximum bus voltages and maximum line currents with an accuracy guarantee verifiable during actual operations. Despite the need to quantify uncertainties, our procedure requires only a few deterministic load flow runs, leading to a significant speedup over standard methods that require a loop to solve at least one optimization problem for each bus and line. Extensive benchmark case studies demonstrate the practical accuracy and efficiency performance of the proposed method. For example, we can conduct interval load flow analysis for a 10,476-bus system achieving a relative error of less than 10% in about 0.5 seconds.