The interplay between climate policy uncertainty and stock market performance has emerged as a pressing research question in light of the challenges posed by climate change to financial markets. This paper measures China's daily and monthly climate policy uncertainty (CPU) from Jan 2000 to Mar 2022 based on Chinese news data for the first time. Then, the nonlinear and lag impacts of the US CPU and China's CPU on the return, volatility, correlation and tail dependence of China's and US stock markets are investigated and compared by adopting copula function and the distribution lag nonlinear model (DLNM). The data of stock markets includes the Shanghai Composite Index (SSCI) and NASDAQ from Jan 2000 to Mar 2022 from the Choice database, and the Shenzhen Composite Index (SCI) and S&P 500 are used for the robustness test. The empirical results indicate that (1) the growth trend of China’s CPU index is similar to that of the US. However, there are significant differences between the impacts of these two CPUs on stock markets. (2) For China, high CPU decreases current stock market return and increases volatility but decreases it in the future. It could also increase the upper tail dependence between China’s and the US stock markets’ volatilities in current period. (3) For the US, CPU decreases stock market return in the short term but increases it in the long term. High CPU increases volatility in short term, decreases volatility in 5 months and increases it again after 6 months. Both low and high CPU could increase the correlation between China's and US stock markets' volatilities.