The answers are not obvious to anyone and economics is a 'rubbery' enough science to permit many points of view. There is no methodology you seek, at best one can identify the value of contracts, resource allocations and so on, and estimate the cost of damage done.

It is also very difficult to compare nations such as Britain and the US which followed orthodox rules of economics, fi8nance and trade, with nations such as the USSR and Germany that simply did 'whatever it took' and where the political regime had complete control over all aspects of resources and labour.

All nations 'printed money' during the war, and took steps to prevent the inflation that almost always accompanies war (as basic like food and fuel become scarce, but demand in the civilian economy remains fairly constant). The totalitarian regimes could simply make bans and closely control markets and distribution. In Western economies, things like 'buy war bonds' campaigns and 'victory gardens' sought to soak up excess money and supplement supplies, with rationing also being central.

Nations sought hard currency where possible. Germany's economy was largely self-contained in the late 1930s, but repeatedly there were crises where weapons, machine tools or chemicals would have to be sold overseas to secure money to buy imports- often precious metals- that Germany herself could not provide. Later, such reasons became justifications for certain military actions.

The US and Britain were trading nations and as such most British purchases of US weapons in the early war years had to be paid for in gold or hard currency, and shipped in British ships (the 'Cash and Carry' scheme favoured by isolationists). The US obtained substantial windfalls as much of Europe was buying anything they could produce, however mediocre, for cash. Many of these weapons deliveries were too late to be of use to their buyers, despite being paid for.

By mid 1940 Britain stood alone, and largely broke with the effort of her rearmament to that point. Yet, the British were keen to place large orders for American weapons. However, US manufacturers were reluctant to gear up for such a massive effort, given the British might collapse either militarily, politically or economically. In such a situation, major US companies would be left with expensive production lines making things nobody wanted.

The Roosevelt government stepped in to both guarantee the British orders and to coerce the manufacturers into making the needed tanks and planes, by financing the production facilities. Over the subsequent 18 months US political isolationism weakened, and soon these facilities were producing weapons for the British, other Allies, and the US Government, with the Roosevelt Administration raising debt to levels hitherto considered grossly irresponsible.

Certainly, much wealth was lost in the nations overrun, or destroyed by fighting. The US did fairly well because of the transfer of wealth, but in particular the establishment of very modern production plant and methodology, combined with a very strong research, development and scientific sector that was largely government controlled. It was this economic integration and depth that served the US so well for future production, while the war gave her a much better-connected nation, provided education and discipline to millions of workers (either as workers or soldiers), all without substantially damaging the nation physically, nor killing very many of her men. It is worth noting that the business sector in the US was predicting ruin as a result of debt levels among other things, every year from 1944 into the 1950s.

This outcome is almost unique in modern history.

The other nations all suffered to a greater or lesser degree, and took much longer to recover. It is a necessarily simplistic notion to say that a line was drawn under the world's economies in 1945, and the prosperity of nations after that was dictated by their ability to produce efficiently, commodities that were needed. This is perhaps not so controversial or surprising. But often the seeming complexity of economics does boil down to such simplicities.

The Wages of Destruction: the Making and Breaking of the Nazi War Economy Adrian Tooze; The Affluent Society JK Galbraith

I'd like to add a little to the posts by /u/pantstime and /u/georgy_k_zhukov, because I think they miss something important.

As the figures from Georgy show, on the whole, the economies of the major powers expanded during the war. (Japan is an exception - I don't know anything about the Japanese wartime economy.) Even while being subjected to naval blockade and massive air raids, Germany managed to grow its economy every year until 1945.

However we should pay some attention to what was being produced. Wartime economies produced what was needed at the time. Tanks, guns, shells, bombers, munitions factories, trained soldiers, etc.. Most of these things have very limited use once the war is over, if they even last that long. Wartime production meets the (relatively) short term needs of the war, but not the longer-term needs of people.

To take a concrete (no pun intended) example, during the war Britain built very little housing, while a significant portion of the housing stock was damaged or destroyed. This left major problems with the supply and quality of housing in the country that had to be made up over the next couple of decades. That's a benign example compared to Germany, which literally used pillage and forced labour (including from its own population) to keep its economy going.

Put simply, economies did generally grow, during the war, but that doesn't mean they were really good for people. Germany might be amazing at producing tanks, bombers and soldiers, but those are only really useful when you're at war, and countries hope to be at peace most of the time. Comparing a wartime and a peacetime economy is, to a degree, comparing apples and oranges.

Which brings us on to the question of what the longer term impact of the war was. Take a look at the chart here(1) which shows a selection of notable economies (note that this is per capita not total GDP - the picture is roughly the same either way though). What is striking is how temporary the impact of the war is. If you look at the long-term trends, by the late 50s most countries are back where you'd expect them to be if the war hadn't happened.

US GDP grows incredibly quickly, then falls almost as much (in absolute terms) as Germany's. If you skipped over the 30s and 40s, you'd imagine a pattern of pretty steady growth. Germany is pretty much back on track by 1960.

Of course, this high-level view hides a lot. The impact of the war is so large that it hides changes in performance of different economies, and we've glossed over deliberate postwar attempts to build a better world economic system and to rebuild European economies. Nevertheless it's striking how quickly economies reverted to something like "normal" levels.(2)

(1) This is from Maddison project data, which is essentially an update to the set Georgy's data is drawn from.

(2) With the proviso that the 1930s wasn't really "normal" for the US - I think it was JK Galbraith who called it a bigger trauma than the war - nor for Germany, which got on to an unsustainable war footing earlier than most.