All of a sudden videos won't play

Some more ideas - Did you previously have external volumes connected, that are no longer connected? Then connect them again and try to play the video again.

Some video formats are container formats, with metadata tags and a resource fork containing the actual video, and the information may be distributed across devices. For example, copies of QuickTime Movies may not include the video component at all, but reference the video in the original, and if the original of the copy with the referenced resource is no longer available, the video cannot play.

A video may also not be able to find the video component, if the format of the disk is incompatible, for example, if it is case-sensitive.

iCloud Photos may not have downloaded the resource, if you are running low on storage, and "Optimise Storage" is enabled. Then make sure, you have plenty of free storage and a working internet connection.

Photos 5 an Catalina can no longer play some older video formats at all. Could you play the problematic videos at all, directly after the Catalina upgrade?

If you could previously play the videos, but no longer after the Mac restarted, check, if your Mac is in Safe Boot Mode. Sometimes a Mac may accidentally come up in Safe Boot Mode, and this disables third-party system extensions and advanced features of the graphics. This can disable some video formats.

Restart the Mac, to be sure it is running in normal mode. See: How to use safe mode on your Mac - Apple Support

Hi

What format is the video, taken on what device?

Do you have "optimise mac storage" selected?

Where is your library stored? On your system drive/in you pictures folder, or somewhere else.

When was the last time you played this particular video (or is it all videos - if so when did you last play a video). Has there been a system update since then?

Is your MacOS version still 10.15? Do you have automatic updates turned on? If so has there been a recent update.

Are you able to export the video by selecting it and then file>export unmodified original?