Phosphorus donor nuclear spins in silicon couple weakly to the environment, making them promising candidates for high-fidelity qubits. The state of a donor nuclear spin qubit can be manipulated and read out using its hyperfine interaction with the electron confined by the donor potential. Here we use a master-equation-based approach to investigate how the backaction from this electron-mediated measurement affects the lifetimes of single and multidonor qubits. We analyze this process as a function of electric and magnetic fields and hyperfine interaction strength. Apart from single nuclear spin flips, we identify an additional measurement-related mechanism, the nuclear spin flip-flop, which is specific to multidonor qubits. Although this flip-flop mechanism reduces qubit lifetimes, we show that it can be effectively suppressed by the hyperfine Stark shift. We show that using atomic precision donor placement and engineered Stark shift, we can minimize the measurement backaction in multidonor qubits, achieving larger nuclear spin lifetimes than single donor qubits.