The play was fairly new, we can assume it started as early as 2 weeks after WAT, seeing how they knew about Combustion Man's death.
Taking that into concideration, i would assume it ran at most 1 month, and at that time, yeah, most people would think "Zuko broke up with Mai, and is in the group where the other girl he likes is, they're probably a couple now" but it wouldn't spread so much, seeing how infamous Zuko and the Gaang was at the time.
Zuko in the finale is Fire Lord and is seen with Mai, and the servants can confirm that they are together, again, Zuko is the Fire Lord, the news would likely spread like fire (no pun intended, well, maybe a little) so yeah, as far as the Fire Nation knows, Zuko is with Mai and may have had a short relationship with Katara.
Please use the Ask A Question thread next time.