Continuing downloads on Steam
There should be a folder called Program Files/Steam/Steamapps/downloading which contains files currently being downloaded. You'll need to start the download on your new computer, pause it, get the files off your old computer (from /downloading), navigate to Program Files/Steam/Steamapps/common/Rainbow Six Siege, and put the files (from the old computer) there (the new computer).
Verify the integrity of your game files in the game's Properties > Local Files window. Then, continue your download. Steam should now pick up from where you left off on the last computer.