|Even though you may have "owned" the files on the old computer, the user name on the new computer is different, even though the name may be the same. The UID connected with each user name will be different and that is what matters.|
That is why you cannot delete a user and then recreate it with the same name and expect to see all the old users files. It doesn't work that way, you need to take ownership of the files with the new user name.
It is certainly not a FAT-NTFS issue because all versions of Windows from XP onwards are capable of reading both NTFS and FAT files system. It makes no difference at all which file system the Operating System is installed on.