Time machine stopped working with Catalina 10.15.5, why?

I am using Time Machine to back up to a Synology NAS.

Ok.. I am surprised as Synology NAS is one of the best.

Both Tesserax and I use Synology NAS and recommend them as best alternative now to the Time Capsule.

Perhaps this is an issue that came with the latest patch. Since it was working ok beforehand and so is your wife's computer on 10..15.4

Do read the info in Synology tutorials about how to setup the NAS for Time Machine.

The advertising bonjour over SMB is important.

https://www.synology.com/en-global/knowledgebase/DSM/tutorial/Backup/How_to_back_up_files_from_Mac_to_Synology_NAS_with_Time_Machine

To force it to use SMB simply uncheck AFP as a protocol that is being used ..

Also you don't have to set SMB level. As long as you did not limit SMB then Catalina will always choose most secure option.

If you really want to fiddle you can.

It may be worthwhile to remove the disk in Time Machine... and reconfigure it exactly according to the Synology tutorial.

To do this you need to turn off Time Machine in Preferences.. then using add and remove backup disks

Then reboot the computer and after following the tutorial add the disk back into Time Machine. This will renew the Time Machine plist configuration.

I would still recommend using Carbon Copy Cloner while you are having issues. It has hugely superior reliability.. if less integrated into Mac OS.

I am being pre-emptive here.. but very similar situation.. issues suddenly appear with 10.15.5 to a Time Capsule.. from the evidence there I would definitely suspect SMB1 is culprit.

https://discussions.apple.com/thread/251499777?answerId=252909767022#252909767022

There is another way to force the connection to AFP protocol and that might help with your particular issue.

Exactly as in previous thread.. use Go Connect to Server in Finder.

Once you have the disk mounted in Finder then you can select it in Time Machine and see if that works better than using SMB.

Sorry, but that is just not enough information in order for us to be able to assist you with a potential solution.

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Is this to a Time Capsule?

Be assured you are one of a multitude.. and AppleDNTCare.

The TC is now using older AFP protocol where Catalina really requires SMB2 or later.. not available on any Apple router.

I suggest you use a third party backup software like Carbon Copy Cloner.

Or if you wish to continue with Time Machine buy a large USB drive and plug into your computer.