While I'm afraid I can't answer your question about which (if any) countries independently developed nuclear weapons, I can answer about an experiment that was conducted in the 1960s that determined how likely it was that another country could develop them without help.

Called the Nth Country Experiment, the question was this: the United States and Soviet Union already have nuclear weapons, but how long would it be before a third country had them? And a fourth? And a fifth? And the nth? Using only the publicly-available information of the mid-60s, could a country go from no nuclear weapons to having not only the expertise to do so, but also determine what infrastructure was required to enrich weapons-grade nuclear materials and develop a thermonuclear bomb?

I'll cut to the chase: the answer is yes, and as long as you have a Ph.D in physics, you can develop the the theory for a bomb in as little as three years.

Three students who had recently received their doctorates in physics fields unrelated to nuclear weapon design were selected by the University of California Livermore and given a simple task: using their knowledge as physicists and the information on fission bombs the United States believed was accessible for a sufficiently-motivated country and nothing else, design a nuclear bomb and describe the steps required to build one.

And three years later, they did. At least theoretically, and here's where things get a bit fuzzy. The bomb as designed by these physicists was never built or tested; instead, their design was reviewed by "an informal committee" whose names, titles, or experience are not made clear in the report, which is unfortunately still highly redacted.

While the gist of the study is clear — they designed a bomb that would be effective and with some experimentation, could be further refined into something very effective. But it did have limitations: the report notes that while the designers may have been over-optimistic in their yield predictions ("They correctly observe that they have very little firm information about the criticality of their system. ... it would seem that confidence in the expected yield is unwarranted.") — but they still managed to design a nuclear bomb from nothing. Later, it's noted that "the calculation of the yield ... is usually the most complex calculation required[.]"

A strong theme throughout the report is the need to test such a device. The authors say that while the "Final Design is credible without a test," they "see no way to design a credible thermonuclear explosive without testing." And such tests wouldn't go unnoticed: a "weapon production complex consists of about 10 buildings on 30-40 acres" and "requires a staff of about 400" — and that's before fuel production, which require the use of a "200-megawatt reactor" to produce 50 kg of plutonium per year, on top of plants for further refining uranium an plutonium into fissile materials.

In a 2003 article, it's pointed out that "access to fissile materials remains the most significant barrier to the proliferation of nuclear weapons" — while you could come up with a design for a bomb in a potentially very brief period of time, the challenge was getting the nuclear components required in such quality and quantity that you could actually execute on that design.

One final note: the article is very concerned with the rate at which China was developing its nuclear program, and for good reason. In 1960, they began rapidly accelerating research and development (with the help of the Soviet Union), and completed their first successful nuclear test in 1964, the same year the Nth Country Experiment began. Just two years later — a year before the Experiment completed — they had produced a nuclear missile that was "'deadly accurate.'"

This is a difficult question to answer since there are two ways that you could look at it. The first is the question "Can every other nations' nuclear program be traced to the Manhattan Project?" and the second is "Could anyone else have conducted a Manhattan Project?".

Can every other nations' nuclear project be traced to the Manhattan Project?

There are five nations that are party to the Treaty for Non-proliferation of Nuclear Weapons that had developed and tested nuclear weapons prior to 1967. Those nations are USA, Great Britain, France, China and USSR (now Russia). This reply focuses on those first.

If you just look at nuclear programs then Great Britain actually had the first one as part of its Tube Alloys project and were the first ones to calculate the critical mass of enriched uranium ie. the mass required to make a bomb. Coupled with discoveries by other scientists (more on this below), the Tube Alloys project made a definite recommendation that a bomb was feasible and could be produced before the end of WWII. Unfortunately, Britain lacked the resources (material, access to uranium and financial solvency) to pursue this avenue of research and the Tube Alloys program languished.

Around the inception (mid 1940) of the council which would eventually kick off the Manhattan Project, Britain offered to share its research on the feasibility of atomic weapons with the USA and recognised that the USA had far greater resources with which to pursue such weapons. The Quebec Agreement (1943) was signed to effectively merge the national programs with the understanding that Britain would receive nuclear technology and that atomic weapons would not be used against targets without British consent.

By the end of the war, British consent having been obtained for the strikes on Japan and after the death of Roosevelt who had signed the Quebec Agreement, the USA did not fulfill its promise to share nuclear technology with Britain. This resulted in Britain having to restart its own national program without explicit American assistance but with the fruits of the cooperation obtained during WWII. Despite efforts to renew the nuclear partnership with the USA, Britain would not achieve a successful test until 1952.

This is overly long for the first section so let me be brief on the other countries:

France's nuclear program did not start until after WWII with the contribution of Bertrand Goldschmid who had worked on the Manhattan Project and had developed a better method for enrichment of plutonium as having worked with Enrico Fermi on the construction of the first nuclear reactor (Chicago Pile 1). France would not test its first device until 1960.

The Russian nuclear program actually began in the mid 1930s (I have not been able to ascertain the date with more accuracy than that, sorry) with a committee to investigate the feasibility. This is directly analogous to the American Advisory Committee on Uranium formed in 1939 and was actually a few years ahead of it. However, the German invasion on what would become the Eastern Front greatly disrupted Russian efforts and little progress was actually made until the tide of battle turned after Stalingrad in 1942/3.

Still, the Russian program was years behind the American one and the recognition of the potential destabilisation of the geopolitical situation that could occur after the Germans were defeated led the Russians to investigate espionage as a way of catching up. Several sympathisers were recruited, notably Klaus Fuchs and Theodore Hall and those men passed several key documents and blueprints across to Russia. The end result is that Russia tested their first nuclear weapon in 1949.

China came late to the development of nuclear weapons and were greatly aided by the USSR in the late 1950s. The Soviets are known to have exported reactor technology, prototypes and advisers to China to assist with their nuclear program and China conducted its first nuclear test in 1964.

Looked at it this way, while the Manhattan project was undeniably the main driver in the development of nuclear weapons, both Britain and Russia got into the game first (so to speak) and the main contribution of the Manhattan Project to their own programs was to allow for avoidance of the sort of criticality accident that led to the deaths of Louis Slotin and Harry Dahglian. In other words, their respective programs were accelerated but not defined by the data obtained (legally or not) from the USA.

The French appear to have had no other assistance except for Bertrand Goldschmid who admittedly worked on the Manhattan Project in the essential area of enrichment of the fissile matter. However, according to your initial query, the French program undeniably stems from the Manhattan Project and was undoubtedly bolstered by the knowledge that a bomb was certainly feasible in actuality.

Finally, China was directly assisted by Russia and definitely did not develop weapons by itself.

So the proximate answer to your query is a resounding "NO". No one developed nuclear weapons independently of the Manhattan Project. However, I am far less familiar with the development of weapons in India, Pakistan and North Korea (all of whom claim to have nuclear bombs) and Israel (who has left their status deliberately ambiguous) and it may be that they achieved their development independently but that is unlikely given that the fundamental design of nuclear weapons was an open secret by the time that they conducted their first tests.

Could anyone else have conducted a Manhattan Project?

Which leads me to the science behind nuclear weapons. The fundamental concept is very simple and Leo Szilard was the first to propose the mechanics of a nuclear chain reaction as early as 1933. The actuality of nuclear fission was discovered by Otto Hahn and Fritz Strassman of Germany while the Tube Alloys project pioneered the technique of gaseous diffusion that was to feature prominently in the Manhattan Project. Various other scientists contributed in a greater and lesser amount to the various discoveries needed to produce a nuclear bomb and those scientists were not limited to those residing within USA.

The possibility of a nuclear bomb was openly discussed amongst physicists during the 1930s and the core mechanisms that would need to be developed were very much common knowledge in people with the requisite background. In other words, "everyone" knew that a bomb was possible and it is quite reasonable to assume that, absent the Manhattan Project, other nations could have and would have developed the bomb.

In particular, even if Russian espionage had failed, they had both the means and the incentive to engage in their own Manhattan Project style endeavour to catch up to the USA as fast as possible. France's own development of weapons with relatively minimal outside assistance is evidence that it was certainly possible to produce a bomb without resorting to espionage.

References:

The Making of the Atomic Bomb - Richard Rhodes

Dark Sun: The Making of the Hydrogen Bomb - Richard Rhodes

In one sense this is practically a philosophical question. One could argue, quite fairly I think, that simply the mere existence of functional nuclear weapons is a great boon to any nuclear bomb making program. It demonstrates quite adroitly that nuclear weapons of significant destructive power can be manufactured by an industrialized nation and be of a design such that they can be feasibly dropped by bomber aircraft of the time (e.g. B29s). This alone significantly constrains the problem space and eliminates a lot of blind alleys and false paths. If your estimates show that it would take an insane amount of resources to produce a bomb, or a bomb would have to be unfeasibly large, you know that you made a mistake somewhere, either a fundamental modeling mistake or some sort of calculation error, because you have evidence to the contrary. Additionally, knowing that the endeavor is feasible and practical could lead some countries to pursue it, whereas if it were merely hypothetical there would be huge open questions of just how feasible and practical it was (even for the US with all its resources and industrial might it was a substantial challenge, imagine undertaking a similar endeavor not knowing if it might be 1-10x as difficult).

Consider this in the context of the German nuclear program, or what passed for one, during WWII. They had made a good number of missteps in their research and modeling, so much so that they thought any nuclear bomb would have to be made of literally many tonnes of fissile material. There are recordings of captured German scientists (the Farm Hall transcripts) discussing the problem after learning of the Americans use of an atomic bomb on Japan. If those scientists had been part of an ongoing weapons program at that time, instead of being PoWs, they could have used the mere fact of a successful atomic bomb to revisit their models and figure out where they went wrong.

Anyway, let's talk about countries that absolutely can trace their weapons program to the Manhattan Project in one way or another. First, let's start with the Soviets. They had a program that started during WWII and they had some pretty competent folks running it, smart enough to figure out that the Americans were working on a bomb project because all the American nuclear scientists suddenly stopped publishing research papers for some reason. However, the Soviet war effort was a much more dire struggle than it was for the US, so they were forced to keep their weapons program on the back burner during the war. Additionally, they gained a huge amount of help from espionage from inside the Manhattan project, likely accelerating their program by several years. It's a pretty safe bet that even without the espionage the Soviets would have built a functional nuclear weapon, but exactly how long it would have taken and how good the initial design would have been are pretty big unknowns.

Then you have the UK. During WWII they were part of the Manhattan Project, and responsible for some very key research. However, in 1946 the US made its nuclear weapons program secret and kicked the UK out. The British then built an entirely homegrown nuclear weapons research and development program, building weapons successfully by the early 1950s, based partly off of some of the Manhattan Project research and their own work.

China's program received significant assistance from the Soviets, so could be said to trace its heritage back to the Manhattan Project as well.

France, perhaps, could be considered to have developed its nuclear weapons independently (as much as is possible anyway). They had a nuclear power program at the start of the war which was pretty well founded. During the occupation the French were able to divert their Uranium and heavy water supplies they had procured for research to Morocco, keeping them out of German hands. After the war the French restarted their research and built a functional reactor in the late 1940s, eventually leading to the production of nuclear power plants and a nuclear weapons program (which reached fruition in 1960).

Later "proliferation" of nuclear weapons technology to Pakistan, India, North Korea, Israel, South Africa, etc. occurred via routes for which determining the exact heritage is tricky and deciding whether or not such programs could trace a line back to the Manhattan Project would be largely an academic exercise. Even more so since a lot of material from the Manhattan Project has been declassified and basically everyone has access to it.

Edit: It's worth adding a notable P.S. here to talk about nuclear technology held by non nuclear powers. There are today many countries who have all of the most difficult and important "pieces" of nuclear weapons development who do not possess any nuclear weapons. For example, Japan has a full-fledged nuclear power industry, is capable of enrichment, Plutonium reprocessing, etc, they could trivially produce a nuclear weapon if they desired. The same is true of other countries such as Brazil, Germany, Canada, the Netherlands, etc. Nth generation proliferation can occur through those programs (as it did through the Netherlands to Pakistan, North Korea, Iran, and Libya via A. Q. Khan). Evaluating the lineage of such proliferation is challenging, you can trace Pakistan's program to the technology of the Zippe centrifuge which itself traces back to the Soviet weapons program, but then do you attribute the whole Soviet weapons program as a descendant of the Manhattan Project (which failed to develop a working centrifugation facility)? It gets complicated fast due to the complex global interdependency of industry.