Unprecedented investments in clean energy technology are required for a net-zero carbon energy system before temperatures breach the Paris Agreement goals. By performing a Monte-Carlo Analysis with the detailed ETSAP-TIAM Integrated Assessment Model and by generating 4000 scenarios of the world's energy system, climate and economy, we find that the uncertainty surrounding technology costs, resource potentials, climate sensitivity and the level of decoupling between energy demands and economic growth influence the efficiency of climate policies and accentuate investment risks in clean energy technologies. Contrary to other studies relying on exploring the uncertainty space via model intercomparison, we find that the CO2 emissions and CO2 prices vary convexly and nonlinearly with the discount rate and climate sensitivity over time. Accounting for this uncertainty is important for designing climate policies and carbon prices to accelerate the transition. In 70% of the scenarios, a 1.5 °C temperature overshoot was within this decade, calling for immediate policy action. Delaying this action by ten years may result in 2 °C mitigation costs being similar to those required to reach the 1.5 °C target if started today, with an immediate peak in emissions, a larger uncertainty in the medium-term horizon and a higher effort for net-zero emissions.