Animal-based preclinical testing is increasingly misaligned with the biological and regulatory demands of modern drug development. Persistent failure of drug candidates underscores the limitations of using animal models for human drug development. In oncology and neurodegeneration, where disease mechanisms and therapeutic responses are largely human-specific, clinical success remains low despite decades of refinement of rodent and xenograft models. Meanwhile, the regulatory framework for human-relevant New Approach Methodologies (NAMs) has become significantly more defined. In addition to the U. S. Food and Drug Administration (FDA) Modernization Act 2.0 and 3.0 and the FDA's 2025 roadmap, the FDA has issued draft guidance on the general use of NAMs in drug development and on streamlined nonclinical programmes for monoclonal antibodies, while the UK Medicines and Healthcare products Regulatory Agency (MHRA) and European Medicines Agency (EMA) have each signalled more formal entry points for non-animal data. Here, we synthesise the emerging NAM ecosystem - engineered 3D cultures, patient-derived organoids, microphysiological systems (organ-on-a-chip), bioprinting, and AI-enabled in silico modelling - and evaluate each platform's position within the drug development pipeline. We argue that bioprinting should be viewed not as a standalone model class but as an enabling biofabrication strategy that can improve the reproducibility, architectural control, and manufacturability of organoids and organ-on-chip systems. Critically, we frame adoption as an engineering-regulatory translation problem: scalable manufacturing, assay robustness, defined biomaterials, and cross-site reproducibility must advance in parallel with biological fidelity. We propose a practical readiness framework and a tiered deployment strategy that enable immediate reductions in animal use in high-impact domains, notably safety testing and oncology triage, while building toward replacing more complex and longitudinal endpoints through integrated multi-organ and computational approaches.