We present ROSA -- Roundabout Optimized Speed Advisory -- a system that combines multi-agent trajectory prediction with coordinated speed guidance for multimodal, mixed traffic at roundabouts. Using a Transformer-based model, ROSA jointly predicts the future trajectories of vehicles and Vulnerable Road Users (VRUs) at roundabouts. Trained for single-step prediction and deployed autoregressively, it generates deterministic outputs, enabling actionable speed advisories. Incorporating motion dynamics, the model achieves high accuracy (ADE: 1.29m, FDE: 2.99m at a five-second prediction horizon), surpassing prior work. Adding route intention further improves performance (ADE: 1.10m, FDE: 2.36m), demonstrating the value of connected vehicle data. Based on predicted conflicts with VRUs and circulating vehicles, ROSA provides real-time, proactive speed advisories for approaching and entering the roundabout. Despite prediction uncertainty, ROSA significantly improves vehicle efficiency and safety, with positive effects even on perceived safety from a VRU perspective. The source code of this work is available under: github.com/urbanAIthi/ROSA.