The restriction imposed worldwide for limiting the spread of coronavirus disease 2019 (COVID-19) globally impacted our lives, decreasing people’s wellbeing, causing increased anxiety, depression, and stress and affecting cognitive functions, such as memory. Recent studies reported decreased working memory (WM) and prospective memory (PM), which are pivotal for the ability to plan and perform future activities. Although the number of studies documenting the COVID-19 effects has recently blossomed, most of them employed self-reported questionnaires as the assessment method. The main aim of our study was to use standardized tests to evaluate WM and PM in a population of young students. A sample of 150 female psychology students was recruited online for the administration of two self-reported questionnaires that investigated psychological wellbeing (DASS-21), prospective, and retrospective memory (PRMQ). Subjects were also administered two standardized tests for WM (PASAT) and PM (MIST). We found increased anxiety, depression, and stress and decreased PM as measured by self-reports. The perceived memory failures agreed with the results from the standardized tests, which demonstrated a decrease in both WM and PM. Thus, COVID-19 restriction has strongly impacted on students’ mental health and memory abilities, leaving an urgent need for psychological and cognitive recovery plans.