You're right that it makes most sense for the the topmost connective under an $\exists$ to be a conjunction (or an atomic formula).
Writing something else under the $\exists$ is not wrong, but it will rarely be the most natural to express the meaning you want.
For example $\exists$ distributes over disjunction, so instead of "there is something that is an apple or an orange" $\exists x.(Ax\lor Ox)$ it is usually more natural to think "there is an apple or there is an orange" $(\exists x.Ax)\lor(\exists x.Ox)$ or equivalently $(\exists x.Ax)\lor(\exists y.Oy)$ -- as can in fact be seen from the English phrasing you suggested for $\exists x.(Ax\lor Ox)$.
In classical logic $p\to q$ is the same as $\neg p\lor q$, so $\exists x.(Nx\to Wx)$ can be said simpler (?) as $(\exists x.\neg Nx)\lor(\exists y.Wy)$ "there is something that is not a nuke, or weapons of mass destruction exist". This makes it clearer that the actual contents of the statement doesn't really depend on whether any particular thing is simultaneously a nuke and a WMD.
Finally $\exists x.\neg p(x)$ is the same as $\neg\forall x.p(x)$. Here it is less clear which wording is the most natural. Sometimes it is most clear to say that there is something that isn't a pony; at other times it is more natural to say that not everything is a pony.
We can use the $\neg$ correspondence to rewrite $(\exists x.\neg Nx)\lor(\exists y.Wy)$ into $\neg(\forall x.Nx)\lor(\exists y.Wy)$, which we can then recognize as equivalent to an implication: $(\forall x.Nx)\to(\exists y.Wy)$. So
There is something such that, if it is a nuke, it is also a WMD.
is actually equivalent to
If all there is are nukes, then there is a WMD somewhere.
The latter formulation seems at least to me to give the more intuitive grasp of what statement claims.