Why won't my computer complete the update? I have tried 3 times.

What error message is presented ?

How exactly was the update 12.5.1 downloaded ?

There are factors:

1 - Like poor Wifi connection

2 - Slow wifi connection

3 - Too many other people downloading at the same time and Apple Severs are overloaded

4 - Software on the computer interfering with the download process including AntiVirus Software

Notation: Disabling the AV Software may or may not solve the issue. Removal of the AV Software may increase the chances of the download to perform as expected.

5 - Running behind a VPN which may increase Packet losses or corruption of the download 

6 - If via wifi too many Other devices using the wifi at the same time reducing the Bandwidth to your computer. 

7 - Best Results are via Ethernet Cable to Router

8 - Restart in Safe Mode. This will perform a Disk Repair, clear cache files and only load Apple Software, extensions and fonts. The boot up will be slow and can take some time - Normal and Try Again using the Software Update Applet in System Preferences 

9 - Worst case - Restart in Recovery Mode and choose Reinstall macOS. This in theory will pull latest version of Monterey available and commence the installation.

Ah, the final minutes are allows a gross misrepresentation of The Real Time it will take - in my experiences since about Big Sur and forwards

The countdown clock says 5 minutes and in Real Time could be up to 30 minutes.

Patience and a second hot beverage of choice and come back and presto 12.5.1 is up and running

Welcome 😀

Do come back to the Apple Support Communities ( ASC )  in the future, if / or when additional questions may arise

The below is predicated on your Internet Service Provider ( ISP ) has stabbed your Account with them and have Installed there Equipment and is tested by their Service Person as Functional

Connect to the internet with your Mac

Same situation, here.

Mac Book pro , 16", Intel

v. 12.5 is installed and working properly

v. 12.5.1 ... cannot be installed - internet connection is stable,

"safe mode" option also doesn't seemed to be a useful technique

for upgrading. When I tried it - it said that the last stable version

appropriate for "safe mode" is v. 12.5

What is happening?

Thanks for the detailed answer. Yes, may be during the first try/attempt of the upgrade,

I was using wi-fi and this caused some downloading errors. It happened often to have

connection problems.

Fortunately, I still have Ethernet Cable somewhere at home. I will try this way.

Keep you in touch with the results.

Okay, Ethernet cable helped! :)

The Upgrade is done!

Of course the last 10mins were more than mentioned 10mins

but at least the upgrade finished correctly and didn't stuck on

15mins for hour.

Thank you

How much free space?

Also, as this is very recent, you may want to give it a few hours to a day for it to propagate everywhere.

Storage space:

1/4 - taken space

3/4 - free space

"you may want to give it a few hours to a day for it to propagate everywhere" - this is strange...

Yes, please do keep use posted on the progress - time permitting.