I don't think I can answer this question in general (I'm not sure it's even possible) but let me try a partial answer.
First, "global influence" comes in many shapes and forms: cultural, scientific, technological, economic, diplomatic, military, intelligence... Naturally there is a large overlap since a country which is advanced in one domain is usually advanced in the other domains as well. Nevertheless nuclear power clearly belongs to the domain of military power, and so far no country has tried to use their nuclear power in order to gain other kind of influence, for instance blackmailing other countries into an advantageous trade deal.
Most countries also agree that nuclear weapons should actually never be used: it is supposed to be only a deterrence weapon, which means that other countries can't even risk an attack on a country which has nuclear power. This is true even for big nuclear powers such as the US and Russia during the Cold War: neither could have afforded to attack the other since both would have been wiped out as a result (this is called the doctrine of mutual assured destruction). It's also worth mentioning that most countries in the world (but not all) have signed an international non-proliferation treaty (NPT) to prevent the spread of nuclear weapons.
As a consequence, nuclear weapons are not exactly a bargaining chip that nuclear powers can easily use to increase their influence. So the only advantage is the status, since countries which have nuclear weapons belong to a very small club:
- US, Russia, UK, France and China officially have nuclear power and are signatories of the NPT.
- India, Pakistan and North Korea officially have nuclear power but are not signatories of the NPT (originally North Korea was but withdrew in 2003).
- Israel is "deliberately ambiguous" about whether it has nuclear power or not and is not a signatory of the NPT.
It's probably impossible to evaluate exactly whether "belonging to the club" really gives these countries more influence, and if yes how much.
To what extent are nuclear weapons necessary to ensure influence on the international stage?
Under a general definition of "global influence", nuclear weapons cannot be said to be a precondition since there are countries which have significant international importance and don't have nuclear weapons (Germany and Japan for instance).
Is there a distinction on this basis between established states, who may be more economically developed, or with more existing diplomatic relationships, and newer states that may see the only insurance policy against regime change instigated by foreign powers as being a credible nuclear threat?
There is a distinction defined in the Non-Proliferation Treaty: it acknowledges the countries which have built and tested a nuclear explosive device before 1967, while the other countries agree not to seek to acquire them:
the NPT non-nuclear-weapon states agree never to acquire nuclear weapons and the NPT nuclear-weapon states in exchange agree to share the benefits of peaceful nuclear technology and to pursue nuclear disarmament aimed at the ultimate elimination of their nuclear arsenals
The case of "rogue countries", namely North Korea and (to some extent) Iran, is problematic not only because they seek to obtain nuclear weapons but also because they behave in a very threatening way. Their real or potential access to nuclear weapons might give them some bargaining power but it also makes them very isolated on the international stage, so it's not clear whether they really gain any global influence.