I agree that it's no; there's nothing there that says the skirmish phase resets just because you change the companion.
Just keep in mind that a skirmish is just like any other phase. It goes like this.
1. Start of the skirmish.
At this point the skirmish does not yet involve Gandalf, so Shadowfax doesn't trigger.2. Skirmish actions.
Gandalf replaces the other companion.3. Skirmish resolved and wounds placed (and overwhelmed characters killed).
4. End of the skirmish.
Note that this can also function the other way around. Here's a more complicated example:
Aragorn has one wound and is "bearing"
Brego and assigned to skirmish a
Desert Spearman. Gandalf's skirmish was just resolved. The Free Peoples player knows that the Shadow player has 2x
Whirling Strike and a
Red Wrath in hand, as well as the twilight to use them.
1. Start of the skirmish. - Mandatory action:
Desert Spearman exerts for
Brego, leaving him with 1 vitality.
- Optional actions: The Free Peoples' player passes; the Shadow player removes
![3 [3]](https://lotrtcgdb.com/forums/Smileys/classic/3.png)
to wound Aragorn (using
Desert Spearman's text).
2. Skirmish actions. - The Free Peoples' player passes
- The Shadow player plays
Whirling Strike, wounding Aragorn, leaving him with 1 vitality (note that he couldn't play
Red Wrath to kill Aragorn because the
Desert Spearman is exhausted from
Brego).
- Worried about Aragorn, the Free Peoples player plays
Mighty Steed.
- The Shadow player plays another
Whirling Strike to wound Gandalf.
3. Skirmish resolved and wounds placed (and overwhelmed characters killed). - Gandalf wins the skirmish and the
Desert Spearman takes a wound, killing him.
4. End of the skirmish.