We use the electronic spin of a single nitrogen-vacancy defect in diamond to observe the real-time evolution of neighboring single nuclear spins under ambient conditions. Using a diamond sample with a natural abundance of ^{13}C isotopes, we first demonstrate high fidelity initialization and single-shot readout of an individual ^{13}C nuclear spin. By including the intrinsic ^{14}N nuclear spin of the nitrogen-vacancy defect in the quantum register, we then report the simultaneous observation of quantum jumps linked to both nuclear spin species, providing an efficient initialization of the two qubits. These results open up new avenues for diamond-based quantum information processing including active feedback in quantum error correction protocols and tests of quantum correlations with solid-state single spins at room temperature.