It would really help if you told us if you are using Lightroom (Lr icon) or Lightroom Classic (LrC icon); and also tell us the version number.
Assuming you are talking about Lightroom Classic:
Basically is it possible to restore backups on a new system if old system suddenly stops working?
What i mean is that i had edited a number of raw pics on the old system but neither did i export to the cloud or export those raw files to jpeg.
Exporting or saving in the cloud is not needed, and not necessary.
Will copying and pasting lets say the backup folder resolve this issue?
Sure if you do it correctly. See: https://www.lightroomqueen.com/restore-backups/