China started pursuing nuclear weapons in the mid-1950s. So a full decade after World War II, and well after Mao had consolidated his power. They reached completion in the mid-1960s — two decades later. So the state of China after World War II, while not entirely irrelevant, is not really the best way to think about China's capabilities at this time.

They had some technical assistance from the Soviet Union, until 1959. The Soviets helped construct enrichment plants, but did not tell the Chinese exactly how everything worked. When they pulled out it created major havoc in the Chinese program. (There was a question about this recently, so I will link to my answer there.)

All in all, it took the Chinese about 10 years to go from the decision to make a bomb to actually detonating one. They devoted huge amounts of resources to the task. They were not "technologically inept" — they were viewed as "technologically backwards" at the time, but as with the Soviet Union, India, Pakistan, North Korea, and lots of other places that have developed nuclear weapons despite having not-entirely-industrialized economies by American standards, it is quite possible for a not-totally-industrialized nation to develop nuclear weapons, and to concentrate areas of very high technology within an overall landscape of low technology. (Which should complicate any simple generalizations about technological "backwardness" and its relationship to weapons proliferation.)

10 years is a long time to work on a bomb. The US did it in about 3-4 years (still the fastest nuclear program ever); the Soviet Union did it in about 4-5 years; the British in about 4-5 years; the French in about 4-5 years (The spans are based on trickiness in pin-pointing the real "start" of a bomb program — I don't consider purely theoretical studies to be the real "beginning" of a bomb program.) So the Chinese spent about twice as long getting their first bomb together as most other states prior to that point. Some of that is arguably because their "assistance" from the Soviet Union was actually counter-productive (see my previous post on this). Some of this is because developing the necessary technical infrastructure was indeed difficult for them at that time (and because some of their political decisions actively inhibited this development), even though a lot of the basic technology had been declassified (reactor technology, much enrichment technology, many basics of weapons design were all declassified in the late 1950s as part of the "Atoms for Peace" program and the attempts to develop a peaceful commercial nuclear power industry).

So it was a very big effort, and a non-trivially difficult one. For the West, the Chinese bomb was a real wake-up call — a sign that any power willing to make sufficient sacrifices could probably pull off a bomb. It is not coincidental that after the 1964 test, the US and USSR both became more interested in developing the Nuclear Non-Proliferation Treaty (which both had been lukewarm towards in the late 1950s, when it was first introduced). The Chinese program is a funny demarcation point. Half of its decisions/history looks like the Western programs that came before it (e.g. focusing on gaseous diffusion, which is a highly-efficient means of enriching uranium but technically among the most difficult, and hardest to hide, as well as their decision to immediately pursue thermonuclear weapons in record time), and the other half looks more like the countries that came after it (e.g. Israel, India, Pakistan, South Africa, North Korea) in that it was initially reliant on direct foreign assistance (which often proves counter-productive), was used to develop theatre-specific weapons (the Chinese arsenal has always been pretty limited in size), and was the product less of their country having reached a sufficient level of "technological maturity" than the state deciding to concentrate immense resources into one technological area.

For furthering reading into the early stages of the Chinese program, check out John Wilson Lewis and Xue Litai, China Builds the Bomb (1988). For somewhat later developments, Jeffery Lewis has just published Paper Tigers: China’s Nuclear Posture, which looks pretty rich and interesting (I have not yet read it, just skimmed some chapters).

Japanese PPP per capita in the period around WWII was only 3-4 times that of China. And while high for an Asian country, Japan's standard if living wasn't actually at western levels until the post war years.

http://en.m.wikipedia.org/wiki/List_of_regions_by_past_GDP_(PPP)per_capita#World_1.E2.80.932003.28Maddison.29

With it's greater population, China theoretically could've been able to match Japan. But China was a weak and unstable regime, having just emerged from the Warlord era in which central author had totally collapsed, and still in the midst of a civil war with the Communists. Japan, on the other hand, had long had an extremely stable, unified government, and had been building up their military might.

And when it came to WWII, the Japanese captured northern China and the important southern port cities. They soon stagnated after that (southern China is mountainous and much more favourable to the defensive side; it is a repeating trope of Chinese history that the north gets quickly swept over by invading forces while the south holds out). However, even though the Chinese could hold out, having no port cities presents obvious supply problems.

By the 1960's, in contrast, the Chinese had a strong, unified government of their own, and were the second most powerful country in the socialist sphere. They were backwards in terms of quality of life, sure, but due to their immense population they were still able to achieve status as a medium power. Their GDP was 2/3 that of the UK, which was easily able to put itself in the nuclear club. The rest was merely a matter of will, they made it a priority to develop the technology and allocated sufficient resources. In the 60's, especially, they had a critical need of it, as they were in a bitter rivalry with the Soviet Union, sometimes at the brink of war, and thus required a nuclear deterrent (obviously, unlike most countries threatened by the USSR, they couldn't just rely on the US to deal with it).

I'm not an expert, and /u/restrictdata has given a more comprehensive answer than I can, but I would suggest not underestimating the advantage of knowledge diffusion from not being first. A couple points:

China benefited from western-educated scientists that played a key role in developing their weapon. Deng Jiaxian was trained at Purdue University. Peng Huanwu had 2 doctorates from Edinburgh University. Guo Yonguai was educated at U of Toronto, California Institute of Technology, and Cornell. And there are a number of other Chinese scientists that were trained in the west. It's difficult to know exactly how instrumental these people were in developing China's actual weapon, but these individuals became senior scientists in the Chinese program and in the development of an education infrastructure for new nuclear weapons scientists.

Also, despite the fact that the Soviet promised help to China (initially offering very substantial help -- plans, mathematical models, even a 'sample' bomb) almost completely fell apart due to overriding geopolitical relationship issues between the two countries -- at least contemporary U.S. thought was that the Chinese gaseous diffusion plant at Lanzhou shared characteristics with Soviet diffusion plants.

Factual source: Spying on the Bomb: American Nuclear Intelligence from Nazi Germany to Iran and North Korea, Richelson (Amazon Link)

You make China seem completely primitive and backwards in your comment, but this is rather silly. You say "massive losses" against the Japanese, but the Chinese only suffered around 3.5 million combat casualties (Killed, Wounded, and Missing) against the Japanese ~2 million combat casualties. The Soviet Union, for comparison, suffered over 10 million combat casualties, and Germany suffered around 3-4 million combat casualties. It should also be noted that China had a much larger population than these other nations. Granted, Chinese civilian casualties were heavy, but so were Soviet civilian casualties, and proportionately these were a much greater amount of the population.

What "did" impact China's ability to develop weapons was the fact that the infrastructure had been almost totally gutted by the war. While the Chinese were able to make small arms, it wasn't that they couldn't afford heavier weapons, it was that their industry had not yet developed the precision works required for heavier weapons and vehicles. Prior to the onset of the 1937 Marco Polo incident, Germany had already provided the Chinese military with significant aid in return for shipments of Chinese raw materials. Chinese arsenals were producing avant-garde German equipment like the iconic Stahlhelm helmet, the MG34, and the Mauser 98k, and some were beginning to produce heavier weapons like field guns and even light vehicles. There was also shipment of aircraft and even several warships (which ultimately never made it to China) in exchange for Chinese currency. So "barely afford to make the most basic weapons" significantly understates China's warmaking potential.

TL;DR: Both talent and industrial base was available, and Chinese enjoyed a period of peace and massive development at time. So making nuke is not hard, just need to make effort into it.

If you can read Chinese, there plenty of resource on this subject, which offers quiet a bit of detail and are primary sources.

A general summary paper on China nuclear project.

Memoir of Chinese nuclear engineer Wei Shijie He was a graduate who's first job was at Research Base 221 (also known as Atomic City, the base was decommissioned in 1993 and handed to local government. now it's just another Chinese city. You can see pictures of it here), and spent the next 30 years of his life on the project. (He also have personal memoir that talk about the everyday life at the secret base. Including what it is date and life with your family with no contact with outside world).

And a excellent documentary on this subject as well, it many interview of projects directors, as well scientist and Engineers that worked on this program as well as many previously unseen footage. As well as how geopolitics around the world affected this problem. (Sino-Soviet split actually accelerated this problem, since 1) there is more urgency. 2) The Soviet aid camp left, so the camp led by western returnee can finally push their plans and request resources. (Two camps are always in conflict, with one side arguing that China must align with Superpower to modernize. While the other side argue that by relying on Soviet aid will eventually make China a Soviet satellite state.）

By the 1950s, people with a decent background have a good idea of how a atomic weapon should be build. It just a matter of have the industrial base to support the effort and ability to build and support a sizable stockpile. The bulk of Chinese R&D is focus on making a missile deployable H-bomb, with the A-bomb development simply been the H-bomb primer. (Which why China's H-bomb came so quickly after the A-bomb, and why the first H-bomb test is actually a bomb been dropped from a plane rather than a nuclear device mounted on a tower)

Talent You can also see this focus when you look at the research focus of China's nuclear scientist. (which there are quiet a few)

http://en.wikipedia.org/wiki/Wang_Ganchang He was known to be able to figure a implosion design that's small enough to be a thermal nuclear primer, but doesn't have as stringent of a manufacturing requirement that's with in reach of China's machine tools.

http://en.wikipedia.org/wiki/Qian_Sanqiang

http://en.wikipedia.org/wiki/He_Zehui

http://en.wikipedia.org/wiki/Deng_Jiaxian

http://en.wikipedia.org/wiki/Yu_Min_(physicist)

http://en.wikipedia.org/wiki/Zhu_Guangya

http://en.wikipedia.org/wiki/Zhou_Guangzhao

http://en.wikipedia.org/wiki/Cheng_Kaijia

http://en.wikipedia.org/wiki/Peng_Huanwu

Another thing is don't underestimate China's potential industrial base. Check out this sites for many nuclear facility are in China during that time.

http://fas.org/nuke/guide/china/facility/nuclear.htm

One unfinished site is out a tourist attraction and the facility is definitely non-trivial even with everything stripped out for tourism.

http://www.gochengdoo.com/en/blog/item/2783/816_chinas_oncetopsecret_underground_nuclear_plant

Just as a side note, at same time as this project, the mid 1950s. China was pursuing nuclear fusion via Stellarators and Laser Inertial Confinement (proposed in 1961 after China's first laser demostration, and started in 1965), which sur There is a nice paper on the Stellarators (which is cancelled in 1966 due to cultural revolution), and the laser inertial confinement project continued to today though full funding only after 1987 allowing them to push beyond technology demonstrators and start building experiments.

Qian Xuesen (one of the founders of the Jet Propulsion Laboratory) might have been the reason.

"In the same year, he finalised plans for what was to become the Dongfeng missile, launched in 1964. Through his work, and with help from the Soviet Union, China made great strides and tested its own nuclear bomb in October 1964."