Applications that have not gone thru Apple's Notarization process can't be run on Catalina and above. According to Apple's information to developers:
Gatekeeper on macOS helps protect users from downloading and installing malicious software by checking for a Developer ID certificate from apps distributed outside the Mac App Store. Make sure to sign any apps, plug-ins, or installer packages that you distribute to let Gatekeeper know they’re safe to install. And now, you can give users even more confidence in your apps running on macOS Mojave by submitting them to Apple to be notarized.
As of this writing, there hasn't been any published way to go around the Notarization process to run applications.
Update: To address concern about older software not running, Apple has also stated this in their Support doc:
Beginning in macOS 10.14.5, all new or updated kernel extensions and all software from developers new to distributing with Developer ID must be notarized in order to run.
Apple recommends that you notarize all of the software that you’ve distributed, including older releases, and even software that doesn’t meet all of these requirements or that is unsigned. Apple’s notary service uses a variety of methods, including telemetry, to determine which of the above rules to relax for preexisting software.
I'm reading that last quote to mean only new software and not existing code. I think we need more clarification from Apple on what happens to existing applications.