Three things used to set up a process to handle incoming SAT-C weather. - Standing orders, night orders and posted chartlets of the NAVAREAs and/or Forecast Areas.
In the standing orders I have a requirement that the most current weather be posted. Weather for outside the area is thrown away by the officer on watch. Once this is done a couple of times it only takes a few seconds to identify which part to keep and which part to dispose of. After this become a routine it's very easy.
The chartlet, copied from List of Radio Signals for the area the ship is in is kept posted also. It's a lot quicker to see if you are, for example "NORTH OF 31 N to 67 N WEST of 35 W" if you can see the area being described on a diagram.
Also in the standing orders is a requirement to call the master for unexpected weather. To keep from being woken up when I expect severe weather I write out in the night order what I expect. For example "expect frontal passage tonight with winds shifting NW to 45 kts". That way I don't get called if that's what happens.