Keyboard Maestro should work fine with Catalina without any appreciable issues.
Note: I will not be updating to Catalina any time soon, and would recommend most people should wait some time before updating.
Keyboard Maestro does not yet know about the rename of iTunes to Music, so the native iTunes actions will not work in Catalina currently, though that should be resolved in 9.0.3 shortly, and in any event the actions are light weight shims on AppleScript, so you can resolve them yourself if you are in a hurry, or using an older version of Keyboard Maestro.
Catalina removes support for 32-bit applications, and so with Catalina, Keyboard Maestro 2.x and 3.x will no longer work, but Keyboard Maestro versions back to 4.x should continue to work in Catalina (though obviously with caveats based on older versions not knowing about modern security issues).
You will have to ensure you enable Accessibility for both Keyboard Maestro and Keyboard Maestro Engine. If you have any troubles with accessibility (eg typing keystrokes, pasting, etc), try toggling the accessibility permissions for Keyboard Maestro and Keyboard Maestro Engine off and then on again.
In particular, Catalina has a bug that may result in the Keyboard Maestro Engine checkbox not toggling when you click it (even after you unlock the System Preferences). If you encounter this, you should report this to Apple using the Feedback mechanism - the more people who report it, the most chance Apple might fix it, though I am not holding my breath.
Thankfully, once you convince it, there should be no further problems. You should be able to remove the Keyboard Maestro Engine entry from the accessibility permissions, then quit and relaunch the Keyboard Maestro Engine (in Keyboard Maestro, File ➤ Quit Engine, File ➤ Launch Engine), and then you should be able to enable it. You probably do not need to restart. See Accessibility Permission Problem assistance if you have any problems with this.
Keyboard Maestro will ask for permission to access your Contacts to fill in a default email address for contact or crash reports, but you do not have to allow that. It will be required if you use any of the AddressBook tokens.
Keyboard Maestro doesn’t ask for permissions for Location Services, Calendar, Reminders, Photos, Camera, Microphone or Full Disk Access, not unless you run some sort of script or action that accesses them (there are no native actions to access them currently, though there may be in the future).