GDP is not, for the most part, an important determinant in whether a country "goes nuclear" or not. Because nukes are expensive, the economic health of the country will play a role in making the decision, but in most cases the countries that have decided to go ahead with making nukes were doing it primarily out of a sense of security, and security often is at the expense of economics. The Soviet Union, for example, did not have an excess of resources at its disposal in the late 1940s, as it was recovering from WWII — its program imposed a huge burden on its economy. The UK was itself not in a financially good position when it decided to build its own nuclear weapons (and tried very hard to economize for that reason). The Chinese economy was not in good shape when it built its own weapons (China was going through both the Sino-Soviet Split and the Cultural Revolution), and neither India, Pakistan, nor North Korea have had economies similar to that of the wartime USA. (I don't really know about France's economy in the late 1950s, so I can't comment, or South Africa's in the 1970s, or Israel in the 1960s, though I cannot imagine the latter having been in any state comparable to the US or USSR.)

It is difficult to isolate the most salient factors for proliferation, though many political scientists have tried to do it over the years in a very quantitative way. For me, the best way to understand it is to know that it takes a definite decision to go down the path of actually producing a weapon (as opposed to just doing some preliminary research, which many nations did), and that the decision will have economic, political, and security implications. What many (but not all!) of these nations did have were pre-existing scientific-technological infrastructures that could be adapted to this task. That is something that the rich oil countries have tended not to have: they lack the human infrastructure, the institutions of science and technology, the sort of thing you cannot simply buy overnight but must build up over decades. (The North Korean program, for all of its black-market dealings, was only really made possible because they started to build up their scientific infrastructure starting in the 1950s.)

Nations that rely very heavily on other nuclear states for their security have turned out to be the least likely to develop their own programs, both because they didn't feel it was necessary (they are under the "nuclear umbrellas" of a host state) and because if they did go down that path, the host state would punish them (the US threatened to abandon Taiwan and South Korea should they pursue independent nuclear weapons, for example, and that proved a sufficient threat). States that have felt "on their own" ("pariah states") have often felt the most incentivized for getting nukes (Israel, South Africa, North Korea) for the security reason, and given their preexisting isolation may have found it (ironically) politically easier than those states who are more fully enmeshed with other economies.

Anyway — this is a complicated question and no single explanation works. If you want to know how India and Pakistan got nuclear weapons, one has to look closely at those examples, and not try to generalize for all. But I hope my answer has given a little bit of the flavor of the explanation for these things — it's complicated.