Green hydrogen and derived electrofuels are attractive replacements for fossil fuels in applications where direct electrification is infeasible. While this makes them crucial for climate neutrality, rapidly scaling up supply is critical and challenging. Here we show that even if electrolysis capacity grows as fast as wind and solar power have done, green hydrogen supply will remain scarce in the short term and uncertain in the long term. Despite initial exponential growth, green hydrogen likely (≥75%) supplies <1% of final energy until 2030 in the European Union and 2035 globally. By 2040, a breakthrough to higher shares is more likely, but large uncertainties prevail with an interquartile range of 3.2–11.2% (EU) and 0.7–3.3% (globally). Both short-term scarcity and long-term uncertainty impede investment in hydrogen end uses and infrastructure, reducing green hydrogen’s potential and jeopardizing climate targets. However, historic analogues suggest that emergency-like policy measures could foster substantially higher growth rates, expediting the breakthrough and increasing the likelihood of future hydrogen availability.