If I were to guess:
The new world order should see the US maintain its dominance, although it'll have to cede Asia increasingly to a rising China, hostile Russia and unreliable Indonesia and India. Plenty of energy to back them too on this thread's topic.
The pecking order will be followed by China - second strongest economically and militarily even today, a country which has ambitions to dominate the next two centuries as the US did as most people know. Plenty of energy to back them too on this thread's topic.
Third should see Russia with its arsenal of weapons, access to gas and uranium and significant land mass. It'll also benefit from the rise of China. Plenty of energy to back them too on this thread's topic.
At fourth we'll see a host of new emerging nations with strong population do what China did 10 years ago - become a manufacturing hub and food bowl, and make lots of money and improve standards of living significantly. These should include Indonesia, Brazil and India as front runners.
Fifth will be the remnants of the old world (these will compete with No #4) and that should simply be Germany and Japan, the two leaders of the old world economically and militarily in Europe and Asia respectively. Competing with them will be the new world's emerging low cost, high tech and low employment countries such as South Korea and Turkey (former Ottoman Empire, don't forget).
I think the biggest losers will be western Europe in the next century. In particular, we will see Italy/France/Spain head towards the dark ages, at least 2 generations lost, and an off chance the UK slips out of top 10 too within our life times. Not surprising, each of these countries had their turn to rule the world within the last 5 centuries (except maybe Italy), and if anyone's been there recently you'll guess they are probably at least 20 years away from getting out of a mess where half the people under 35 are not employed and have zero hope of finding employment. Much worse than Japan for all practical intents and purposes. Despite the mounting debt in Japan, at least the country is high-tech, efficient and people have jobs, just not stable ones. You can't even get a casual job in Paris/Barcelona these days.