If you’re trying to argue that the UK has a right (and indeed a duty, since you’re talking about “responsibility”) to have nuclear weapons based on those points, then there are a whole heap of countries that should be getting tooled up.
Germany and Brazil have bigger economies than the UK (and Italy between the UK and Russia, another nuclear state – makes sense to add Canada in there too as they’re next on the list). The UK used to be 4th biggest, but has slipped to 7th, and will probably continue to do so. Unless we’re talking about passing nukes around to whoever the top five are at any one point, then this is a flawed premise.
21 countries have higher populations than the UK, including Egypt, Bangladesh, Vietnam, Ethiopia, DR Congo and Iran as well as the afore-mentioned states of Germany and Japan, with Italy again right behind the UK. Overall there are about 35 countries with “above average” populations (the average being around 34 million), including Myanmar, Kenya and Tanzania.
In the globalised economy we now find ourselves in, very few countries (developed ones, certainly) could say they didn’t have global economic and territorial interests. Same goes for treaty commitments. This is a complete non-argument that seems completely ignorant to the realities of the 21st century (just like nukes themselves!)
As for “projecting power” internationally, aside from the horrible picture it paints of one country trying to bully others and the flawed idea that “might is right” that it underpins, this sounds like nothing more than an excuse for high military spending and for having a nuke itself. It’s like saying the boy with the big baseball bat deserves to keep the big baseball bat because he projects power… with his big baseball bat. I’d also be interested to know which of the other nuclear powers – USA, China, Russia, France – is incapable of doing this (and which one is only just capable), and therefore should get rid of their nuke immediately.
If the UK didn’t already have a nuke, there is no credible argument for why it should tool up ahead of countries like Germany, Japan, Italy, Australia, Brazil or Canada.
And that’s without even getting into the non-NPT nuclear states of India, Pakistan and North Korea – none of whom would really fit into your rules for a country being worthy of having a nuke – or Israel and their “we don’t have a nuke, honest” stance.