I think I have solved it for myself! Starting from nothing more than the linear dependence of U,V,W.
Since they are linearly dependant we can write:
W=a*U+b*V where a and b are constants. Being a 4vector equation this is true in any frame.
If we transform to the rest frame of U then in this frame w is parallel to v by W=a*U+b*V, u=0,0,0. Now we rotate the spatial axes so that x is in the same direction as w (and hence v).
Finally performing a standard LT with x axis defined in this way, does not change the direction of v or w and causes u to transform from its rest frame into a frame in which it is parallel to x and hence v and w.
