The established nuclear powers don't present a huge amount of threat because it has been established that nuclear weapons will never be used by them unless provoked by the most dire circumstances (such as Mutually Assured Destruction where they are used in response to an imminent nuclear threat).
However, a lot of emphasis is put on preventing new nuclear powers from rising (i.e. North Korea, Iran, etc.) because it is somewhat uncertain how this newfound power would be wielded.
To answer the question more directly, I think that a country establishing itself as a new nuclear power would certainly generate a lot of attention and perhaps cause their demands to be heard more loudly, but should those weapons ever be used against a country allied with a recognized nuclear state, it would likely result in their own destruction (at least in part).