This study presents the development and testing of a conversational speech system designed for robots to detect speech biomarkers indicative of cognitive impairments in people living with dementia (PLwD). The system integrates a backend Python WebSocket server and a central core module with a large language model (LLM) fine-tuned for dementia to process user input and generate robotic conversation responses in real-time in less than 1.5 seconds. The frontend user interface, a Progressive Web App (PWA), displays information and biomarker score graphs on a smartphone in real-time to human users (PLwD, caregivers, clinicians). Six speech biomarkers based on the existing literature - Altered Grammar, Pragmatic Impairments, Anomia, Disrupted Turn-Taking, Slurred Pronunciation, and Prosody Changes - were developed for the robot conversation system using two datasets, one that included conversations of PLwD with a human clinician (DementiaBank dataset) and one that included conversations of PLwD with a robot (Indiana dataset). We also created a composite speech biomarker that combined all six individual biomarkers into a single score. The speech system's performance was first evaluated on the DementiaBank dataset showing moderate correlation with MMSE scores, with the composite biomarker score outperforming individual biomarkers. Analysis of the Indiana dataset revealed higher and more variable biomarker scores, suggesting potential differences due to study populations (e.g. severity of dementia) and the conversational scenario (human-robot conversations are different from human-human). The findings underscore the need for further research on the impact of conversational scenarios on speech biomarkers and the potential clinical applications of robotic speech systems.