Any suspicious, unexpected or unexplained death in a hospital- general or mental health- would be reported to both the police and coroner. It is the decision of the coroner whether an inquest is warranted. The police would become involved if foul play was suspected. There would also be an internal investigation by the hospital managers.
youre almost right. No inquest if a doctor writes a cause of death certificate. Without that, automatically an inquest. Coroner doesn’t have a choice about that.