How do I fix a MacBook that won't start after updating to macOS Catalina?

You neglected to tell us the exact model of your Mac. You can get this information by entering the system serial number on the check coverage page here (please don't post the serial number on the forum since it is considered personal information):

Check Your Service and Support Coverage - Apple Support

Upgrading to Catalina implies this is an older computer which may contain a Hard Drive as opposed to an SSD. If this is the case, then that Hard Drive may have failed. No....the upgrade did not break the hard drive, but just put an already failing drive over the edge.

The other possibility is you have some third party software installed which is interfering with the normal operation of macOS. Try booting into Safe Mode which should bypass most issues with third party software interfering with system booting.