Im not a techy but when I changed computers and need to play WoW on the new computer I just copies allthe appropriate folders to an external hard drive and then copied them back onto the new computer. Id be very surprised if you couldn't do the same thing for this game.
If you just install the launcher you download from your accountpage (BUT DON'T START IT YET!). You should be able to copy the "Elder Scrolls Online" folder in your documents (back to the same location on your new computer as well as the "The Elder Scrolls Online EU"/"The Elder Scrolls Online US" folder in your Program Files\Zenimax Online directory.
If you then start the launcher it should work without downloading anything. It worked before for me