How do I retrieve my saves from the cloud?

Solution 1:

Your mid-mission saves are not stored in the cloud. They are stored locally on your computer's harddrive. The Battle.net blog post Campaigning on the Go* explains how Star Craft 2's campaign progress cloud storage works.

The post details the following about what is stored online:

If you’re logging into StarCraft II on a different computer and have previously progressed through a portion of the single-player campaign, you’ll be able to select Continue Campaign and pick up right after the last mission you completed. Campaign progress is stored for you on Battle.net as part of your character profile and includes all missions completed, tech and research upgrade choices made, mercenaries purchased, and Achievements unlocked prior to the most recent mission completion or branching mission choice made.

Below is a compiled screenshot of how you can replay completed missions from Wings of Liberty:

As far as what is stored locally on your machine:

... mid-mission save files. If you want to revert back to a specific save point in the campaign on a computer different from the one you used to originally save your progress, you’ll need to physically or virtually transport your save files. This should only be necessary if you wish to continue the campaign on a separate computer, but do not have Internet access and must choose the Play Offline option. These files are generally located within the StarCraft II folder in the Documents directory of most operating systems. You can copy your campaign save files from within the Accounts folder to a portable storage device, or upload them via email in order to transport as needed.

So there's a definitive answer if you still had access to your save files you could transfer them between machines.

_{* Currently the blog post has a WordPress error so read Google's cached version.}

Solution 2:

Per this thread, saved games are not stored in the cloud - although as you've noticed Campaign progress is saved in the cloud.

Your save files are kept in banks on your computer, so they don't transfer from one computer to the next, but your campaign progress (what missions you have completed before, what achievements you have, whether the campaign has been completed or not, etc.) are on the Blizzard servers somewhere.

If you had the actual save files on another computer, you could move them between computers manually as described here - although unfortunately that does not seem to be the case in your circumstance.