Nuclear spins were among the first physical platforms to be considered for quantum information processing^1,2, because of their exceptional quantum coherence³ and atomic-scale footprint. However, their full potential for quantum computing has not yet been realized, owing to the lack of methods with which to link nuclear qubits within a scalable device combined with multi-qubit operations with sufficient fidelity to sustain fault-tolerant quantum computation. Here we demonstrate universal quantum logic operations using a pair of ion-implanted ³¹P donor nuclei in a silicon nanoelectronic device. A nuclear two-qubit controlled-Z gate is obtained by imparting a geometric phase to a shared electron spin⁴, and used to prepare entangled Bell states with fidelities up to 94.2(2.7)%. The quantum operations are precisely characterized using gate set tomography (GST)⁵, yielding one-qubit average gate fidelities up to 99.95(2)%, two-qubit average gate fidelity of 99.37(11)% and two-qubit preparation/measurement fidelities of 98.95(4)%. These three metrics indicate that nuclear spins in silicon are approaching the performance demanded in fault-tolerant quantum processors⁶. We then demonstrate entanglement between the two nuclei and the shared electron by producing a Greenberger–Horne–Zeilinger three-qubit state with 92.5(1.0)% fidelity. Because electron spin qubits in semiconductors can be further coupled to other electrons^7–9 or physically shuttled across different locations^10,11, these results establish a viable route for scalable quantum information processing using donor nuclear and electron spins.