Control command spacebar should still open the Character viewer even with the latest Catalina update.
In the Messages and Mail apps, you can also see the Character Viewer popup when you click.
How to use emoji, accents, and symbols on your Mac - Apple ...
If this is not working for you—
try restarting your Mac.
To trouble shoot further you can:
Try a SafeBoot https://support.apple.com/en-us/HT201262
Takes noticeable longer to get to the login screen, does a 5-15 minute disk repair before it fully boots up, and certain system caches get cleared and rebuilt, including dynamic loader cache, etc. Login and test. Reboot as normal.
This test will tell you if third party interference; extensions etc are not loaded in safe boot mode.
Test issue in another user (or guest user) account https://support.apple.com/en-us/HT204443
This will tell you if it a universal issue or isolated to your user/admin account.