I've been itching for a while to post a few thoughts here. I have another post rattling around in my head, but I think I'll leave it to incubate a bit longer. I do however have a quick thought I want to get out.
I apologize if I'm misreading this, but it seems to me that most people posting here seem to make the implicit assumption that it will still be government that will govern us. I'm not sure if your use of the word "government" is loosely defined as some centralized form of organization with authority (however derived) to issue orders, organize collective action, and otherwise get people to do things, or you rather mean something related to the nation-state. However, in the opening post, it seems to me that the latter definition of government is preferred. Most of the "forms of government" defined in the wiki link at the top of the thread have been largely developed to apply to the nation-state as a form of human organization, with its various flavors and nuances. Whether we are talking about a liberal democracy, a dictatorship, monarchy (with its many flavors), we are talking about forms of organization that take the nation-state as given.
The nation-state is a relatively new thing in the history of humanity. We generally consider the Treaty of Westphalia as the official birth of the nation-state, but it wasn't until the middle of the 19th century that the nation-state began to spread throughout Europe, and it wasn't until after WWI that it became the only accepted form of organization. Nation-states share a number of basic characteristics: a defined territory, with recognized boarders, a government that has the monopoly of force within the territory, and (some argue) a government generally recognized to have the right to exercise said force within said territory. Today, if some territory is not part of a nation-state (e.g. Palestine, N. Pakistan) we really don't know what to do with it. We give them names meant to show that they fell short somewhere and require fixing of some sort. For example, these regions may be called semi-autonomous regions, or failed states, or no-man's land.
The reason I bring this up is because I want to point out that the nation-state, with all its flavors of internal organization, is ultimately a product of the Industrial Revolution, and its spread and success coincides with the discovery of oil. It blew out of the water all other forms of human organization: city states, clans, tribes, guilds, princedoms, kingdoms, feuds, dominions, lands, and.... no-man's lands, which were a heck of a lot more common before the advent of the nation state. There was a lot of diversity in terms of human organizations back in the day, and the nation-state drove them all to extinction.
That being said, under moderate rates of climate change, I find it hard to see how governments, of the nation-state variety, can maintain their monopoly of force within their domain and fight off external challenges to their sovereignty in the years to come. Political scientists are already wringing their hands over the demise of the nation state in the face of global forces. States find it hard to retain control over what is going on within their territory, the outcomes of their policies are entirely at the mercy of what is happening half-way across the globe. They also find it hard to insulate themselves from external influences, particularly when those external factors could easily undo all the best policies they've put in place over the last century and that people have come to rely on. In the past two decades or so there has been a marked tendency of nation-states to give up pieces of their sovereignty, either to territorial subdivisions within their boarders or to transfer pieces of it to inter-governmental organizations, the EU. If we look outside the OECD nations, we see a whole lot of troubled states, struggling to hold on to some semblance of control - the Arab Spring and its aftermath being the most obvious example.
It is hard to see what could reverse this trend, even under normal circumstances. It is easy to see how climate change, even at moderate levels of impact, would only accelerate and exacerbate it. For example, governments will find it difficult to retain the loyalty of the armed forces when it has a tough time feeding its population. It will be hard for any government to retain the loyalty (loosely defined) of the elites when it cannot protect them from natural disasters or the throngs of people laying claim to their stuff. When resources dwindle, so does the ability of governments to control what's happening within their territory. They will certainly try to retain control, but it is largely pointless: the stuff that created the complexity that allowed governments to exist is gone, and so are nation-states as a form of organization.
That does not mean it's chaos all the way down.
I'll stop here, for now.