If you still have the old machine, go to Start | Run, and type regedit.exe . Regedit will show you Windows' registry and everything that's saved inside it.
Browse to the key named HKEY_LOCAL_MACHINE\SOFTWARE\Cerious Software Inc.\ThumbsPlus\7.0 and make a note of the values of Usercode, Username and Usercount (character case, spaces, punctuation etc. are all critical).
Then enter those same values during installation of v7 on your new laptop, or enter them afterwards through Help | Enter Registration Info.