Traditionally the download preferences were stored in the incomplete folder in files called Download.dat & download.bak. The newest versions of LW appear not use those files so I have no idea where it stores the information. Possibly in the LW preferences folder.
So previously all you did was install LW on new computer, then close the program. Replace the new incomplete folder with the old one from other computer, then restart LW. It is massively important that you only do this whilst LW is CLOSED, otherwise the old download preferences will be wiped.