Which version of LW is on the older computer? The download history is stored in a file called Downloads.dat, though you may not see the extension depending on how you have your computer system view options set.

For LW 4 - 4.16, the Downloads.dat & .bak files were stored in the incomplete folder. In the LW 4.18 & 5 versions, Downloads.dat file is stored in the LW preferences folder.

If you are newly set up on the new computer, you could replace the new downloads.dat file with the old one (but only whilst LW is closed!) This will of course then ignore any newer downloads you have (no history of previous download history for them.) O.o

One possibility, and this is complex & no guarantee of working is to edit your new Downloads.dat file & add the information from the old downloads.dat file but with the new location.

It can be opened with WordPad if you choose open All kinds of documents. But would need to change extension back if saving. I've no idea whether this can work, and you should definitely make a backup of the newer Downloads.dat file before even considering trying.

I've opened the Downloads.dat file on both mac & windows. Windows version looks more complicated lol

Mac example:
w___q~sq~tA/Volumes/Video Art/Incomplete/T-14475015-07 - That Neck.mp3w/xq~sq~wsq~Noˇˇsq~Nˇˇ|sq~Nä*ÔÑsq~Nõˇˇåsq~Nªˇˇ†sq~N «ˇˇ¿sq~N”ˇˇÃxq~sq~w+)urn:sha1:AFW3ONPYSVBG2HH7KU2P VSCBYV7EGFTRq~xq~q~q~t07 - That Neck.mp3q~sq~w

On windows it shows with several boxes near the beginning. Obviously some kind of text encoding my windows system is not enabled to see those characters properly:

___q~ sq~tqC:\Documents and Settings\Administrator\Incomplete\T-5816511-King - Tak - Dream.mp3w\xq~sq~wsq~
×Ïxq~sq~w+)urn:sha1:IABDGS7ILMJ5PZ3UMCVL7ZCJUBNBSW Y7q~xq~q~q~t4King - Tak - Dream.mp3q~sq~w?@sq~!sq~?@w q~%tbpush-endpoint:w4JQTXlFNTBFOFgxNkHGQjhFMjY0OTg1QUU1ODIxQ kUzNzAwNDtmd3QvNTszMzA1NDo5Mi46MS4xMjguMzY=q~(sq~* pq~dpq~-q~eq~/q~2q~3sq~5XÀ¿q~8t
<?xml version="1.0"?><audios xsi:noNamespaceSchemaLocation="http://www.limewire.com/schemas/audio.xsd"><audio title="Dream" artist="King" album="Tak" genre="Progressive Rock" track="13" year="1995" seconds="290" bitrate="160"/></audios>q~;q~2q~=tLIMEq~@q~Bq~Cq~Fq~Gsq~5˜q~Jsq~E
xq~Nuq~Oሯ¸âd˜Zå‚ãpq~Qq~Sxxq~Tsq~tiC:\Documents and Settings\Administrator\My Documents\King - Tak - Dream.mp3w\xq~Wsq~?@wq~Ysq~Zw#A¥ø¡xxq~\sq~5XÀ¿xsq~ sq~?@w

The idea would be to add from the old downloads.dat list to the newer one and make sure the incomplete folder location is identical. I have no idea which of those numbers represents the downloads %. The Download Hash numbers are obvious though (represents all the specific file's qualities & size etc.)