Migrating Mail from Yosemite to Catalina through a Time Machine backup, nothing migrated

As far as I know, actually keeping the attachments associated with the "manual" method I suggest wouldn't happen either.

Since you said that only "at the moment" you don't have access to the original computer, if it's still an option at all I would suggest upgrading that OS X Yosemite 10.10 to something newer (like 10.11 El Capitan, and even 10.13 High Sierra if the hardware supports it, if not further) then making a Time Machine backup off of that and trying to migrate/restore off that.

Beyond that, I think you'd be looking at trying out some 3rd-party data recovery tools like EaseUS, Stellar, Disk Drill, etc. on the V3 mail structure to see if it can keep that together properly maintaining the hierarchy and associated attachments.

Hello Frédéric le Moustique,

Welcome to Apple Support Communities. We understand that you're having problems restoring your mailbox using a Time Machine backup.

Are you trying to restore an IMAP or POP account? Also, are you experiencing this behavior with one or multiple mailboxes?

We've located these articles that goes into detail about that:

Restore items backed up with Time Machine on Mac

Mail User Guide

Hope this helps. Kind regards. 

Hi Frédéric le Moustique,

Thanks for getting back to us with those additional details. It looks like you're having issues restoring old emails from 2 POP email accounts using a time Machine backup.

Just to clarify, do you have another Time Machine backup you can try to restore from/ We're asking as there might be an issue with this specific backup.

If that didn't work, we'd recommend reaching out to Apple Support for further assistance.

Thank you for using Apple Support Communities. Have a nice day.

If you drill down in the ~/Library/Mail/V3 folder I assume if you go to the various mailbox folders (Inbox, Sent, etc.) and drill down further does your Time Machine have all the content there? In other words, did the content as least make it to Time Machine (especially important since you no longer have the old Mac, and since they are POP and don't exist on the server anymore presumably).

The mail folder structure is separated by accounts with a folder name of a GUID (i.e. 1bcd1fb3a-3525c-3627156-162512356c) so even if these are POP emails from an email service you no longer have you can transplant them to an active account you actually have set up.

You can sort out those individual emails files and say, put them on your Desktop, then drag them into your current Mail App structure to import them. If you have dozens of Mailbox folders in a complex hierarchy this would be cumbersome but doable, if you have a fairly simple hierarchy in your Inbox this manual process wouldn't be too time consuming.

If you've confirmed Time Machine actually has the content you could also try importing the Mac user account from Time Machine using Migration Assistant to see if that has better luck handling it than Time Machine did.