Not sure about the timeout of the LED, but you can do this:
PSR-500 Expert Settings: PGM + FUNC + F3 (GLOB); Press FUNC-SEL to see menu help text
Setting Type Menu Name Menu Help Text* Default Values Win500 Name PSREdit Name Description/Notes
Weather Settings WX Check Int Weather 1050 chk interval in SCAN and MAN modes 60 5-250, x100 ms Check interval Tone Detect Threshold
The interval of time to check for a Weather Alarm signal while scanning (WX Priority)
Weather Settings WX 1050Hz Th Weather 1050Hz qualify time 20 7-250, x10 ms 1050Hz qualifier Tone Detect Timeout Length of time to wait on the Weather frequency to check for the Alarm signal
Weather Settings WX Alrm Time WX SAME alert time to unmute 0=unmute @1050 0 0-600, x1 sec Alarm timeout Alarm Settings Length Length of time to wait before unmuting the Weather frequency after an Alarm