I did the point releases (MacOS 12.4 & iPadOS 15.5) and Universal Control works more smoothly for me. Didn't stop working or anything.
If (on the Mac, in Preferences -> Displays -> Add Display (dropdown) you do NOT see
Link keyboard & mouse
[name of your iPad goes here]
... then your systems aren't close enough, aren't both running Bluetooth, don't have WiFi turned on, aren't both logged into the same iCloud Apple ID (not a "managed ID"), or otherwise do not meet the complete set of requirements for Universal Control. I.e., the System/Preference settings must all be correct, etc. as described in the iPadOS 15.5 User Manual
Use a keyboard and mouse or trackpad across your Mac and iPad with Universal Control - Apple Support
Or, as we've also found: any firewall that's turned on must allow incoming connections for at least "Universal Control.app" and rapportd (the firewall itself can even be in "stealth mode" as long as it allows those incoming connections.
Since (with MacOS 12.4 & iPadOS 15.5) Universal Control is no longer Beta, you should be able to get support from Apple Support now. [You can start by going to the top right-hand corner of this web page and clicking on "Get Support"!]