Time Machine Not Backing Up - Cannot Connect to Back-up Disk

Let's start with getting a bit more information about your setup:

• Are you using Time Machine (TM) to back up to an AirPort Extreme base station with an attached USB drive or to a Time Capsule's (TC) internal drive? Your post appears to indicate the former.
• Which exact version is your base station and which version of firmware is it running?
• Which exact version of macOS is your Mac running for these TM backups?
• Is your base station the only wireless router in your current network configuration? If not, what are the makes & models of your other networking hardware and how are they all configured to work together?
• Does your base station's name included special characters or spaces?
• Do you have, as a minimum, IPv6 Link-local only mode enabled on your base station and Mac?
• Finally, what have you tried, so far yourself, to resolve this?

We'll assume that you have already tried basic troubleshooting steps like powering off the AirPort and restarting it again and restarting your Mac, correct?

I've been backing up to AirPort Extreme for a couple of years..

An AirPort Extreme does not have an internal hard drive, so you have either connected a USB drive to the AirPort Extreme or you meant to say that you have a Time Capsule, which is a WiFi router with a built in hard drive designed to store backups made using the Time Machine application.

I do not want to say there was a problem with the last OS Update - but Time Machine has not been able to back-up since then.

Although the release notes for OS 10.15.5 make no mention of anything to do with Time Machine, there have been numerous reports from users who are having difficultly with backups since the last operating system update. It's hard to know how many users are affected by this though, since Apple Support tells me that they are unaware of any issues regarding Time Machine in the latest 10.15.5 Catalina update.

Unfortunately the error code messages are worthless, since Apple does not provide any documentation that would tell us what the error code might really mean.

Nine times out of ten, the "fix" is to erase the Time Capsule drive and start over with a new backup because the current backup file has become corrupted, so not sure whether you want to try a few things that probably won't work before you erase the Time Capsule (or delete the backup of the Mac in question if there are also other Macs backing up.)

If you do want to try a few things.......

Hold down the option key on your Mac while you click on the Time Machine "clock" icon at the top of the Mac's screen

Select Verify Backups

If the process will run, it might take anywhere from 10-60 minutes or more for the check to complete. Let things run until you see that the process has completed or until you see another error message that the process could not be completed

If you want to check on the progress of the procedure once it has begun, hold down the option key again and click on the Time Machine "clock" icon again

Since the backup will not run, the next check will probably be of no value, but it's worth a try.

Hold down the Shift key on your Mac while you click on the Time Machine "clock" icon and select Back Up with Consistency Scan

Let things run until the process completes or until you see another error message.

Report on your results, so we'll have an idea of where to go next. There are more checks, but they are unfortunately not likely to help.

When you are desperate, the old turn it off and then turn it back on method will fix about half of all issues about half of the time.

So, even though it lacks sophistication, it's always worth the time to try this first, rather than last.