In order to transfer your PS4 save games for use on the PS5 you will need to transfer the save using the in-game save transfer option:
1. Load up the PS4 version of No Man's Sky (If using a PS5 see Sony's guide on Switching between PS5 and PS4 versions on your PlayStation® 5)
2. Go to the "Select Save Game" screen
3. Select "Transfer Save Data to PlayStation(R)5"
4. Select "Upload All Save Data"
5. Wait for notification the upload has completed
6. Now launch the PS5 version of No Man's Sky
7. On the Game Mode selection screen, select the cloud button found below the game modes
8. You should now see your PS4 save with an option to download it. Select "Download All Dave Data"
9. Download progress will be shown on screen
10. Wait for the success notification
11. Your save will now appear on the Select Save Game screen and you can continue your progress