You don’t have to use Gnome on Cachy or Fedora. Fedora has spins for nearly every DE, and Cachy also has an option for nearly every DE on install.

PrivacyGuides has a section about this on their website: https://www.privacyguides.org/en/health-and-wellness/#menstrual-cycle-tracking

At the time of writing, it seems Drip is the best option on both iOS and Android. The app is free, open-source, and all data is completely local. Honorable mention to Euki as well