All applications cannot open unless Internet is off
This issue has now been resolved by Apple. The original post is preserved below.

As of 8 pm UTC, Apple's notary service is having 'performance' issues according to Apple's status page, which may be slow enough performance to cause timeouts or simply lock macOS trying to communicate with the service.

Developer ID Notary Service - Performance
Today, 8:00 PM - ongoing
Some users are affected
Users may be experiencing issues with the service. We are working to resolve the problem.

Currently lots of people are installing the just-released macOS Big Sur and putting a lot of stress on Apple's servers, which may have been the cause of this issue.
Catalina's Gatekeeper verifies notarisation on every app when launching the app. This process requires confirmation from Apple before launching the app.
People on slow network connections have suffered from slow app launch since this was introduced. With Apple's servers slow due to the stress caused by everyone updating, the slowness has been moved to Apple's side, but it's still present, and exaggerated for most people.
With the notary service having issues, macOS is unable to contact this service. Since macOS detects a network connection, it keeps trying to communicate with the notary service, rather than giving up as it does when not connected to a network at all. When not connected to the network, macOS defers the check to next time the network is available and may still show a warning when launching apps.
A workaround is to disable AMFI (Apple Mobile File Integrity). This should be considered a last resort, as this disables a key security check on macOS. First, disable System Integrity Protection (csrutil disable from Recovery) to be able to set NVRAM values, then run
sudo nvram boot-args=&quot;amfi_get_out_of_my_way=0x1&quot;

Don't forget to set this back at some point. Reset NVRAM by holding cmd+alt+P+R on boot until the second chime (or just that key with boot-args=&quot;&quot;) and re-enable SIP.