Lost emails

About a year ago, when my Mac updated, the emails from one of the email servers which feeds into my Mac mail application vanished, which also meant I lost all the emails as the settings automatically detected from the online server as well. I wasn't that bothered at the time. However I now need some files which were attachments on those emails.

My Macs have been backed up with time machine going back to 2016, but for some reason the Time Machine isn't giving me the options as on the back up files the 2016-2020 is under my old computer, name Mac Book Air, and even the first back up on this new (M1 MBP) is under a different computer file name for some reason. So the Time Machine gives me no options.

Thus either I need a way to find the mail folder on the backup files, on my external HD, or a way for Time Machine to recognise the previous backups.

Any ideas?