Socially assistive robots (SAR) hold significant potential to assist older adults and people with dementia in human engagement and clinical contexts by supporting mental health and independence at home. While SAR research has recently experienced prolific growth, long-term trust, clinical translation and patient benefit remain immature. Affective human-robot interactions are unresolved and the deployment of robots with conversational abilities is fundamental for robustness and humanrobot engagement. In this paper, we review the state of the art within the past two decades, design trends, and current applications of conversational affective SAR for ageing and dementia support. A horizon scanning of AI voice technology for healthcare, including ubiquitous smart speakers, is further introduced to address current gaps inhibiting home use. We discuss the role of user-centred approaches in the design of voice systems, including the capacity to handle communication breakdowns for effective use by target populations. We summarise the state of development in interactions using speech and natural language processing, which forms a baseline for longitudinal health monitoring and cognitive assessment. Drawing from this foundation, we identify open challenges and propose future directions to advance conversational affective social robots for: 1) user engagement, 2) deployment in real-world settings, and 3) clinical translation.