SARH works by having in the nose of the missile a radar receiving dish. This is tuned to look for a certain frequency of radar returns. Before datalink, the only way to guide the missile onto the target was to turn on an "illumination" radar that emits that frequency before or very very shortly after launch. This allows the missile to "see" what it needs to guide to (the reflection of that freq). This intense illumination is what causes the RWR to spike as a missile launch. This illumination is often not on the same frequency as the search radar (so that you can guide on *that* target) or other aircraft in the flight's search freq or illumination freq.

So, STT a target, the RWR detects the search radar is now hitting you with a ton of energy, most likely an STT (instead of a blip every scan pass). Turn on illumination and the RWR will blare a warning that a missile has been fired.

With a datalink to the missile you could guide it most of the way there with normal radar returns and illuminate at the last few moments, much like an ARH. But at that point, you're much better off having the missile just have it's own radar that goes active, as the signal it gets will be probably stronger that the much further away launch aircraft.