(1) Background: While in many countries, the psychiatric and mental health sectors had been in crisis for years, the onset of a novel coronavirus pandemic impacted their structures, organizations, and professionals worldwide. (2) Methods: To document the early impacts of the COVID-19 health crisis on psychiatry and mental health sectors, a systematic review of the international literature published in 2020 was conducted in PubMed (MEDLINE), Cairn.info, and SantéPsy (Ascodocpsy) databases. (3) Results: After applying inclusion and exclusion criteria, 72 articles from scientific journals were selected, including papers documenting the early impact of the COVID-19 pandemic on the organization of psychiatric care delivery, work processes in psychiatry and mental health units, and personal experiences of mental health professionals. This review identified the contributions aimed at preventing the onset of mental disorders in the early stages of the health crisis. It lists the organizational changes that have been implemented in the first place to ensure continuity of psychiatric care while reducing the risk of SARS-CoV-2 transmission. It questions the evolution of the rights and duties of mental health professionals in the first months of the pandemic. (4) Discussion and conclusions: Although this literature review exclusively documented the early impacts of the COVID-19 health crisis, it is of significant interest, as it pictures the unprecedent situation in which psychiatry and mental health care professionals found themselves in the first stages of the pandemic. This work is a preliminary step of a study to be conducted with mental health professionals on an international scale—the Psy-GIPO2C project—based on more than 15 group interviews, 30 individual interviews, and 2000 questionnaires. The final aim of this study is to formulate concrete recommendations for decision-makers to improve work in psychiatry and mental health.