recover deleted time machine backup

<< I'm on Mojave and backup was made on Big Sur. Could that cause any problems? >>

YES. the drive format was changed in Big Sur, and Mojave can't understand that a Big Sur APFS Volume may have Time machine backups on it -- it is expecting MacOS extended Volume.

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Edit: It is possible you are looking at the NEW Time machine Volume, and that you need to get Time machine to look at the OLD MacOS extended Volume that should still hold your old Backups.

https://www.macworld.com/article/3600278/macos-big-sur-supports-time-machine-on-apfs-formatted-drives-but-there-are-a-few-catches.html

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I was trying to give you a heads-up that the Time machine backups you need may be on your drive (provided you did not deliberately ERASE that drive) but not immediately available -- they are not the most recent.

this older article may also have some applicability:

.Pondini's Time Machine - Troubleshooting -- E2. I can't see some backups

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Sounds good.

¿are the backups showing now the ones you need?

Excellent.

So it looks like the Big Sur upgrade did not DELETE the backups you needed.

But it may have made them hard-to-access.