mailboxes no longer there -how to get them back?

They are all there. They may be hidden. The way mailboxes are displayed changed slightly with Big Sur.

Where are these mailboxes supposed to be? In some of your accounts (IMAP, Exchange) or saved directly in your Mac?

Locate "On My Mac" in the Mail window Sidebar. When the cursor is near the words "On My Mac" you will see a little arrow like ">"

Click to twirl open and show the mailboxes.

Similarly, for mailboxes that are synced to your online accounts, like iCloud, Gmail, etc.

I believe you found those files with extension .mbox somewhere under the V8 folder, right?

Make a new folder, for example in your Desktop - let's call it "things" - and COPY (do not move) all the .mbox folders to this "things" folder.

In Mail, go to

File->Import Mailboxes...

A window will open. Under "Import from", choose "Files in mbox format".

Click Continue.

You will be asked with selecting the location from which to import. Navigate to your Desktop, select the "things" folder you just created, and click Choose.

You will see a list (possibly a huge list) of all the mailboxes in there. Check them all, or pick the ones you want, then click Import.

Sorry - but should has illustrated the suggestion.

refer to below image and input the suggested ~/Library/Mail/V8 the preceding ~ tilde is very important for it to work

Time Machine DOES backup e-mail.

Unfortunately, it cannot easily be retrieved in the way one would like - by navigating back in the Mail interface as we can in the Finder.

We'd have to know the proper folders where mail is stored, and restore these folders.

However, this would lose all the most recent mail, so a tricky business.

Did you connect to the server via a webmail interface to confirm if the mailboxes are really gone from there?

Have a look in ~/Library/Mail/V7

Notation on above, do not remember if for Big Sur the V7 above should be V7 or V8 but maybe the folder are there but not visible

To be fair and balanced - Kudos belong to colleague @ Luis Sequeira1 👍 who has done the Real Heavy Lifting on this issue.

That aside - you are welcome and thank you too 🇨🇦

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

Use favorite mailboxes in Mail on Mac

I am very glad that we could figure it out, together!