Many countries around the world suspect that Iran is attempting to build a nuclear weapon. Regardless of whether this is true or not, is there any objective reason why the US, UK, France, and others are "allowed" to have nuclear weapons, and others cannot, or is the only reason because they are either western allies or too powerful for the US to subjugate? E.g., would neutral countries have any reason to support the possession of nuclear weapons by the US, but object to nuclear weapons by Iran?
The reason why some countries can have nuclear weapons is easy to find in this article: Treaty on the Non-Proliferation of Nuclear Weapons, where you can read very interesting topics.
The Treaty on the Non-Proliferation of Nuclear Weapons, commonly known as the Non-Proliferation Treaty or NPT, is a landmark international treaty whose objective is to prevent the spread of nuclear weapons and weapons technology, to promote cooperation in the peaceful uses of nuclear energy and to further the goal of achieving nuclear disarmament and general and complete disarmament.
First pillar: non-proliferation
Five states are recognized by the Non-Proliferation Treaty as nuclear weapon states (NWS): China (signed 1992), France (1992), the Soviet Union (1968; obligations and rights now assumed by the Russian Federation), the United Kingdom (1968), and the United States (1968) (The United States, UK, and the Soviet Union were the only states openly possessing such weapons among the original ratifiers of the treaty, which entered into force in 1970). These five nations are also the five permanent members of the United Nations Security Council.
Non parties to the treaty
Four non-parties to the treaty are known or believed to possess nuclear weapons: India, Pakistan and North Korea have openly tested and declared that they possess nuclear weapons, while Israel has had a policy of opacity regarding its own nuclear weapons program. North Korea acceded to the treaty in 1985, but never came into compliance, and announced its withdrawal in 2003.
So as you can see in the previous quote there are exceptions to the rule like US, France, China and Russia, and there are countries which haven't approved the treaty like North Korea and India, but the countries that approved the treaty must fallow its pillars.
In fact there is no reason, in theory all the countries have the same rights, and so if United States, Israel, India and others have nuclear weapons then any other country should be allowed to have.
By my point of view if a country doesn't want that other country have nuclear weapons or biological weapons that country must show the example and do what it request and disable all its weapons of this kind, because the rule should be followed by everyone and not only for some unprivileged, because they are equals in rights.
There is really no "good" answer. The countries only have as much rights as they have the actual practical power to enforce given objective worldwide geopolitical realities.
Remember that even UN resolutions don't have any practical power outside UN member states sending military forces to enforce them, or implement economic sanctions effectively.
This is the only objective reason. Nobody has the power to stop US/USSR/China/UK/France from having nukes, whether they would like to or not.
OTOH, the Big Countries on the block sometimes have the power to stop other countries from having nukes - and note that they are not usually successful because they don't have THAT much power (witness Iran - despite all the posturing, the effective rate of success of everyone else in stopping - or even slowing down - their nuclear program is slightly below that of stopping Russians from drinking alcohol during their Prohibition in 1980s).
Pretty much the ONLY examples of countries that were seriously determined to get nukes but were stopped successfully due to non-related circumstances were Syria and Iraq (South Africa and Lybia don't count as they abandoned their programs for unrelated reasons).
 - Not quite - there is also domestic political pressure. A country may refrain from developing or having nuclear weapons because its domestic political considerations oppose the idea for whatever reasons - e.g., Japan opposes it because of the memory of 1945, Ukraine opposed it because they were enticed by material support offers from the West post-USSR-breakup (which I suppose they greatly regretted by now). There are powerful anti-nuclear movements in many western countries.
A second objective reason is that once one country has a lot of nukes, it may be in the interests of stability for other countries without means to match them to have a counter-balancing global power to have nukes. Mutual assured destruction may seem counter-intuitive, but it DID work.
Yes. A very factual reason. Some States have won the last world war, and some others have not.
The current system is indeed unfair to some extent, but pragmatically very effective as ensuring stability. If every state had nuclear weapons then the chance of nuclear war occurring would dramatically increase. If everyone got rid of their nuclear weapons would be that any state could secretly build nuclear weapons use them to bully other states around. The current solution limits both the power of states with nuclear weapons and the chances of nuclear war. The five states recognised as nuclear powers by the treaty may have disagreements, but are exceedingly unlikely to go to war with one another.