The velocity composition formula may yield well-defined values of $w$ for some pairs of $(u,v)$ but that doesn't mean that these values of speeds $(u,v)$ are physically allowed.
Matter and genuine information cannot move by speeds $v\gt c$ because the special relativity postulates – and it is experimentally confirmed – that all inertial frames that are uniformly moving relatively to each other experience the same laws of physics. The Lorentz transformation that is known to map one inertial frame to another is easily proven to map motion at $v\gt c$ to motion backwards in time, at least in some frames.
In other words, using your favoritecomposition formula, $w$ has the right sign and is well-behaved for $uv\lt c^2$ which means that velocities $v\gt c$ seem to sensible composition of velocities in other reference frames that are moving by speeds $u$ obeying $uv\lt c^2$. However, aside from this frame $u$, there exists another frame with a different speed $U$ such that $Uv\gt c^2$ – well, $U$ is sufficiently close to $c$. And in these $U$ frames, the composition formula gives you a reversed sign for $w$ because the denominator goes negative. This is a sign that the chronological ordering of some events – beginning and end of the motion by the speed $v$ – is reversed according to these $U$ frames.
If the object moving with the speed $v$ is a deadly bullet, the observer associated with frame $U$ will see that the bullet kills the target before the bullet leaves (or returns to) the gun. So the cause no longer precedes its effect, and it is a logical contradiction that would allow one to kill his own grandfather before he met the grandmother.
Special relativity demands that such paradoxes are avoided not only in the frame $u$ where the speeds behave well but in all references frames, including the frame $U$ where $1-Uv/c^2$ is negaetive and signals the reversal of the chronology.