Download update but it doesn’t do anything.

...now you can go to the system/update and install a downloaded update  If you want it... It didn't automatically, if it switched off. 

what software version do you have on the watch, are you in the beta program?

Same here. 

I am guessing that you are on the beta firmware 20.23?

It is a bug.

You can always check if there is a new download available. If not it will say "No download available" and that is not a bug.

a "Download Complete" but no updates is a bug.

Yesterday, I checked for updates, and the watch said 'downloading update,' but once it was downloaded, it didn't do anything. In the afternoon and today, I checked again, and it said 'No updates available'.

It was probably an EPO file update.

I have also experienced this before. I believe Garmin uses a Canary Deployment strategy. They are rolling out the release file, but it initially installs only for a small subset of users, i.e., 20% of the users. After a monitoring period, the rest of the users can also install the new version.

Sending out FWs to a specific user/devices is maintained by Garmin. Depending of the deployment of FW you can send a request from your watch for a new FW. If you are in the loop it gets downloaded. IF not you get a message with "No download available".