Almost never...
Note that there is an isometric embedding $X\to W_p(X)$, so $X$ has to be CAT(κ). Second the space $W_p(X)$ contains symmetric $p$-product $S^n(X)=X^{\times n}/S_n$ so $p=2$, or $X$ is one a point-space. Now if $\dim X>1$, then you get into trouble with extending geodesic thru a $\delta$-measure in $S^2(X)$, so you get $\dim X\le1$.