What happens to my campaign progress if I uninstall StarCraft 2?

Campaign progress is stored online. Swapping computers (or reinstalling) shouldn't affect that.

Now that I think about it, I'm not sure if you can save a game in the middle of a mission like you could in SC1, and if so, I'm not sure if that would be saved. But completed missions are completed on other computers.

Update: I have noticed some things that just don't get saved. Like if you finish a mission, then just purchase armory upgrades, then quit. It doesn't always save the armory upgrades. When the campaign is continued, it picks back up immediately after finishing the previous level. But this happens even on the same computer, and isn't any worse if you reinstall StarCraft II

Blizzard on Starcraft 2 Save Games: Starcraft 2 Blog