Gamma-ray emission is observed coincident in position to the evolved, composite supernova remnant (SNR) B0453–685. Prior multiwavelength investigations of the region indicate that the pulsar wind nebula (PWN) within the SNR is the most likely origin for the observed gamma rays, with a possible pulsar contribution that becomes significant at energies below E ∼ 5 GeV. Constraints on the PWN hard X-ray spectrum are important for the most accurate broadband representation of PWN emission and determining the presence of a gamma-ray pulsar component. The results of Parkes radio and Nuclear Spectroscopic Telescope Array (NuSTAR) X-ray observations are presented on PWN B0453–685. We perform a search for the central pulsar in the new Parkes radio data, finding an upper limit of 12 μJy. A pulsation search in the new NuSTAR observations additionally provides a 3σ upper limit on the hard X-ray pulsed fraction of 56%. The PWN is best characterized with a photon index ΓX = 1.91 ± 0.20 in the 3–78 keV NuSTAR data, and the results are incorporated into existing broadband models. Last, we characterize a serendipitous source detected by Chandra and NuSTAR that is considered a new high-mass X-ray binary candidate.