I haven't played in a while
Deepending on how loong ago this was, you might have played in flash (old UI). If you have stuff here :
https://www.warzone.com/Test/LocalStorage it is worth going over to
https://www.warzone.com/ChangeWebMode and see if Flash has the game saved. {the key should look something like warlight_Player_69603_SavedGame_1112765_h } Note that UJS and Unity saves are not listed on that link i think.
That being said, Getting Strategic is not too bad, so you could just play it from the beginning.
What did I do wrong? Do I need to stay on the same computer to pick up my saved game?
Yep. That is how local storage works. And all SingelPlayer in-progress games are saved that way. Deleting cache etc. also could delete your saved games.
Edited 9/25/2018 15:25:40