I point to you the Hossbach memorandum to say: they were not able to build such an army, but they did so anyway. The Hossbach Conference was a meeting between Hitler and some of the top dogs in the military establishment to determine how in God's name Germany was gonna get out of the mess they created for themselves. Rearmanent had essentially sidelined the civilian economy to shove virtually everything into building a huge army. It worked fantastically, but as the 1940s loomed the Nazi Reich was in danger of imminent economic collapse.

Since disarming the military was not an option, Hitler instead outlined a plan to seize industry and arable land in the east, the Lebensraum you're familiar with, to use it to stabilize and hopefully boon Germany's economic trajectory. All present at the Hossbach Conference (including Hitler!) expressed doubts as to the viability of this adventure east, something I think should be shouted from the rooftops: the Nazi leadership thrust their nation into a war they thought they'd lose because they were fundamentally incompetent at governance. This 'efficient Nazi' myth needs to die. Hopefully I'm not getting preachy or moralistic here, I just want to stress how absurd this whole situation was.

Now, this economic weakness was no good for the Nazi Party. Nazi Germany was desperate to cement itself as the 'law and order' government, returning normalcy to a country wracked by revolutionary and economic chaos. They utterly failed to recover the economic situation beyond the general bounce back Germany had experienced since 1933ish, before the Nazi seizure of power. So, they had to divert attention away from how weak the economic structure was. Employing as many people as possible in the war effort was a good start, since less unemployed = less awareness of just how bad the civilian economy is, but it didn't really cover up that the funds simply weren't there.

They focused on community building exercises that masked the depressed economy. German wages never recovered to their pre-depression heights, so what did the Nazis do? Well, the Nazi-run labor union, the German Labor Front, launched a 'Strength Through Joy' program, offering Germans free vacations resorts like Prora or on custom built cruiseships like the Wilhelm Gustolf. The Volkswagen was launched for this purpose too; Germans paid an affordable amount for a brand new car! Sound too good to be true? It was! All the money went to the military and no one ever got a Volkswagen. Sorry suckers.

Propaganda blasted that the Nazi order was granting the people luxury and prosperity and pointed to these programs as proof. Please ignore that you still cannot afford groceries. After all, Goering needs the state funds to buy morphine and pet tigers.

So...if there was no solid economic foundation, how did Nazi Germany pay for massive rearmament? Conquest, fraud and neglect!

I'll give you a solid example of how wacky the Nazi war economy was. Hjalamar Schacht, President of the Reichsbank the brains behind the German 'economic miracle' (which was largely a Nazi myth, but Schacht did great work), devised MEFO bills, which are a great example of this fraud. The Versailles treaty stipulated Germany was barred from a significant investment in war materials. Even if they were allowed, they simply did not have the money in reserve to afford it.

To solve this, Schacht set up a shell organization that printed 'MEFO Bills', and these essentially let the Nazi government pay for arms production without leaving a paper trail. These were, basically 'IOUs' that the government handed to war producers, who could later exchange these for currency. In case this sounds like an unsustainable scheme... you're right. All it was was a way for Germany to run a massive deficit without getting caught red-handed they were spending all their money on war materials. This money needed paid back, Germany never significantly invested in civilian industry, the only way to pay it back was to seize foreign capital. So, war.

Schacht, mind you, was fired in 1939 because he was deeply opposed to how badly rearmanent was destabilizing the economy. Probably one of the brightest minds to ever work in the Nazi government, and he gets sacked and replaced by a total crank, Walther Funk, because he realized Nazi policy was utterly unsustainable.

Jewish businesses were ransacked and captured industry in Czechoslovakia would help re-arm Germany too, but for the most part the German war and civilian economy was incredibly weak, disorganized, and inefficient. Nazi Germany never really streamlined its war production the way the US and USSR would, and in fact did not even engage in total mobilization of the war economy (out of fear that the house of cards would come tumbling down and a repeat of November 1918 would occur) until 1943...after Stalingrad. Yikes.

As to why those other countries couldn't match them: they could, and did. Mobilization began as Nazi's poked the Anglo-French bear repeatedly, but kicked into high-gear after the Munich Conference and subsequent invasion of Czechoslovakia, when it became clear Hitler was insatiable.

You're right that mobilization was not as forced as Germany's, but this is due to the fact France and Britain were shellshocked democratic societies trying to avoid a second war; they couldn't lie and hide mobilization like the Nazi dictatorship, and their government did not thirst for warfare like Hitler did.

Full-fledged mobilization would've been viewed as directed against Germany and heightened international tensions, and many anti-war activists in the West would have viewed such a move as unnecessarily aggressive on the Allied part. So, mobilization was delayed until the last minute, which is part of the reason Czechoslovakia was sacrificed to the Reich.

Still, despite their tepid mobilization, France and Germany were on par; for example, the French had about 4,000 tanks (a good number were obsolete but so were a good number of German tanks); Germany had about 3,000. France had about a million men in the field with another few million mobilizing or in reserves, Germany had about the same. Though Britain had a smaller force, its strategy was to send an elite force to France to hold the line while it mobilized a much larger army when the war began.

In fact, hopping back to the Hossbach Memorandum, the Nazi leadership realized that French and British mobilization would permanently eclipse them by about 1942, and the USSR not too soon after (The Red Army was on paper much stronger than the Germans, though it lacked trained officers and combat-ready equipment; Stalin himself expected the USSR to be fully armed and mobilized by about 43-44)

So, the war was launched when it happened, in the East and West both, as sort of pre-emptive strikes to destroy Germany's enemies while they were still weak enough to be competitive. This didn't work out, but for all the criticism Hitler gets for invading the West and Russia, it really was the only shot he had for Germany to defeat her neighbors before they grew too powerful. Still, we know today he never really had a chance to beat all three European great powers, and throwing the USA into the mix only sealed his fate extra...seally.

The Fall of France is a whole other story on its own, but Germany did not have an inherent material superiority here. They had a more flexible, determined, and confident command structure that was able to outmaneuver and decisively defeat the convoluted, nervous, indecisive Allied command. It was not quality of equipment or quality of troops, but a combination of good leadership and risk-taking on the German side, and poor communication and hesitation on the Allied side, that led to the German victory.

If you want a fantastic, comprehensive, easily digestible overview of what went wrong in France, check out Indy Neidell's two-part video series on this here and part two here This team is just absolutely incredible at what they do and they deserve more love. They're pretty solid historians, especially for what's essentially a pop history channel.

Hopefully this answers your question to some extent :P