What happens if I use Area Zero Underdepths to replace Sky Field, or vice versa? Does either player need to discard Pokemon because the Sky Field/Area Zero Underdepths stadium left play, or is the transition instantaneous?

As long as you meet the required criteria of the new stadium you would not have to discard any Benched Pokemon since one stadium's effect is replacing the other simultaneously.

Relevant cards

SV7_131