How much does climate change & the decline of liberal democracy indirectly increase the probability of an x-risk?

Well, you can look at the recent past to see the impacts of government administrations on policies relevant to climate change and people's well-being. Here are a few points:

• during Trump's presidency, migrant flows to the US suffered a bump. There was some speculation that the flows had a climate link. Whether or not we assume so, I think it's clear that the administration at the time meant to discourage the migrants from seeking refuge in the United States. This has implications for climate refugees in the future. I'm also in doubt about how the US will handle massive internal migration, frankly.
  Handling of migrant flows in Europe were also not encouraging. Key players included conservative government figures and isolationist and racist social movements.
• Brazil's president Bolsonaro adopted aggressive development policies in the Amazon. I've seen a suggestion that development of the Amazon (through burning) is a driver of impacts synergistic with climate change, that is, turning the Amazon into a GHG source, and taking it from rainforest to savannah. In particular, development in the east will interrupt the production and carrying of moisture toward the west, increasing chances of drought over the remaining extent of the Amazon, and thus leading to rapid declines in the geographic extent of the Amazon. This will have impacts on Brazil but also the wider world, resulting in some turmoil.
• Again, thinking of the Trump presidency, but actually, looking at most Republican presidents  since 1980, there's been some friction with efforts to protect the US public from pollution or environmental impacts (really, digging deeper, from all administrations, but you're asking about a particular type of leader). Presence of conservatives in the Supreme Court will impact decisions about carbon taxes, and the legal authority of the EPA to limit corporate activity, for example. I can see this leading to frustration of efforts to mitigate or adapt to climate change (for example, efforts to protect remaining US waterways). Access to clean water is an important need for the happiness and well-being of US citizens.