Several studies have reported that the COVID-19 pandemic has had deleterious effects on sleep quality and mood, but the mechanisms underlying these effects are not clearly understood. Recently, it has been shown that the acceptance component of mindfulness reduces anxiety, and, in turn, lower anxiety improves sleep quality. The purpose of this cross-sectional study was to assess changes in mindfulness traits, sleep–wake quality, and general distress, before, during, and after the first COVID-19 wave, testing the model in which acceptance influences sleep through anxiety in each period. A total of 250 participants were recruited before (Pre-Lockdown group: 69 participants, 29 females, 33.04 ± 12.94 years), during (Lockdown group: 78 participants, 59 females, 29.174 ± 8.50 years), and after (After-Lockdown group: 103 participants, 86 females, 30.29 ± 9.46 years) the first Italian lockdown. In each group, self-report questionnaires, assessing mindfulness facets, distress, and sleep–wake quality, were administered and assessed. The Lockdown group reported lower acceptance and higher depression, while the After-Lockdown group reported lower sleep–wake quality and higher anxiety. The results of the path analysis confirmed that higher acceptance reduced anxiety and higher anxiety decreased sleep–wake quality in all groups. Our results confirm that acceptance influences sleep through the mediating role of anxiety.