Time Machine back up is stuck on Preparing backup... any suggestions?

Have you performed any recent updates on your Mac? If yes, Time Machine has to take a look at everything again....very slowly....before it will start to make a new backup.

Hopefully, you have your Mac connected to the backup drive using a wired connection via Ethernet or USB. If not, wireless on average will take at least 2-3 times longer.

Let the process run. One user who posted recently had to let his Mac run 36+ hours before it actually started to backup, which took another 6-8 hours.

Hopefully, that "software" program was not something like a "cleaner", "optimizer", or "keeper" which will totally mess up Time Machine. Anti-virus applications will do the same thing.

If a database is tied to Lightroom, or an image library, then technically everything in the database or library has likely changed due to the update, so Time Machine will make a new backup of the database, or your image library.

If that is the case, which I suspect, then the backup is going to take some time depending on the size of your image library.....and the application itself.

My advice would be to leave it running unless it stops by itself and provides an error message.

Hopefully, you are not using wireless for the backup.

The problem with stopping at 36 hours is that it might have only required another hour or two to start the backup. If you cancel, then you have to start all over again.

The question really is whether the Mac is connected by Ethernet cable directly to one of the LAN <--> ports on the Time Capsule.

No, unfortunately you will have to decide whether you want cancel the current process, and start over again with a wired connection.