Medical systematic reviews are central to evidence-based medicine, but they remain slow, labor-intensive, and difficult to maintain under the full Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) workflow. Recent LLM-based deep research agents offer a promising route to addressing this challenge, yet reliable deployment in medical systematic reviews remains limited by insufficient clinical domain knowledge and inconsistent adherence to evidence-based methodological standards across the full workflow. We address these gaps with MedSR-Copilot, a PRISMA-aligned multi-agent copilot that decomposes review automation into literature retrieval, coarse-to-fine screening, data extraction, Risk-of-Bias assessment, and evidence synthesis, while preserving structured intermediate artifacts throughout the workflow. We further introduce MedSR-Bench, an end-to-end benchmark for evaluating systems beyond isolated subtasks, from review input to final evidence-synthesis conclusions. MedSR-Copilot completes medical systematic reviews end-to-end under the full PRISMA workflow, achieving 63.6% human-aligned conclusions, 18.3 percentage points above the best baseline among strong general-purpose LLMs and prior automated review systems. In a human-AI collaboration study involving 23 analysis groups across four systematic review topics, MedSR-Copilot, used as a copilot, reduces end-to-end review time by 64.9% and improves final conclusion accuracy by 27.4 percentage points compared with routine-practice workflows. Together, these results demonstrate the reliability and efficiency of MedSR-Copilot as a medical research copilot and suggest a practical path toward trustworthy review automation.