Here in the definition of weighted colimit, it seems to me that the r.h.s.
$C(W⋅F,c)≅Set^{J^{op}}(W,C(F−,c))$
has a wrong direction for the application of the Yoneda lemma.In fact, the Yoneda lemma considers natural transformations from the hom-functor into another functor, but here it is the opposite way!
Indeed, you can't use Yoneda in that spot; but $C(W\cdot F,\sim)\to Set^{J^{op}}(W,C(F-,\sim))$ is the right way around to use Yoneda. If you look at the formula that follows the mention of Yoneda on the nLab page, $$W(j)\to C(F(j),W\cdot F)$$ you'll see that this is precisely signature of the morphism associated with $id_{W\cdot F}$.
Edit: After thinking about this for a few minutes, perhaps your confusion is that you think the isomorphism $$C(W\cdot F,c)\cong Set^{J^{op}}(W,C(F-,c))$$ is intended to be something like the isomorphism $$W(j)\cong Set^{J^{op}}(J(-,j),W).$$ This is not the case. The claim is not that the former isomorphism is an instance of Yoneda; if it were, weighted colimits would be uninteresting because they would be an intrinsic part of every category. Rather, the existence of the former is the substantial fact that a particular set-valued functor is representable; its existence has nothing to do with Yoneda. The instance of Yoneda invoked in the nLab article is rather $$Set^{J^{op}}(W,C(F-,W\cdot F))\cong Set^C(C(W\cdot F,\sim),Set^{J^{op}}(W,C(F-,\sim))),$$ which is used to state that there is a universal arrow $\phi$ from $W$ to $$C(F-,\sim):C\to Set^{J^{op}}.$$
Notice that this is entirely analogous to the isomorphism $$C(\mathrm{colim}\;F,-)\cong C^D(F,\Delta-)$$ (where $F:D\to C$ and $\Delta:C\to C^D$ is the diagonal functor), where Yoneda tells us that the existence of this isomorphism gives rise to a unique $\xi:F\to\Delta(\mathrm{colim}\; F)$--i.e., the colimiting cone.