Did you try switching profiles on the second computer?

On the first computer, did you exit the game and then restart it, so Steam would upload the save? In the save/load list is it shown with a solid cloud symbol (with a tooltip saying it has been synced)?

On the second computer is cloud support enabled in Steam (in the client, open the Steam menu and select Settings, then Cloud) and the game (in the library right click the game, select properties and switch to the Updates tab)?