A search is performed for delayed and nonpointing photons originating from the displaced decay of a neutral long-lived particle (LLP). The analysis uses the full run 2 dataset of proton-proton collisions delivered by the LHC at a center-of-mass energy of s=13 TeV between 2015 and 2018 and recorded by the ATLAS detector, corresponding to an integrated luminosity of 139 fb-1. The capabilities of the ATLAS electromagnetic calorimeter are exploited to precisely measure the arrival times and trajectories of photons. The results are interpreted in a scenario where the LLPs are pair produced in exotic decays of the 125 GeV Higgs boson, and each LLP subsequently decays into a photon and a particle that escapes direct detection, giving rise to missing transverse momentum. No significant excess is observed above the expectation due to Standard Model background processes. The results are used to set upper limits on the branching ratio of the exotic decay of the Higgs boson. A model-independent limit is also set on the production of photons with large values of displacement and time delay.