[Important!] Never Uninstall an EA Game! EVER

Yes, after redeeming your key once, it's no longer valid, but that's not necessary. Your product codes are tied to your Origin account and whenever you log in to the Origin client with the Origin account that you've used to redeem the code, you will be able to see all your purchased games under the "My Games" tab. From there you can always install them again no matter if you are at a new computer or you've previously uninstalled it.

Like adventbahamut already said, make sure that you are logged in with the correct Origin account and your games will be there for you under the My Games tab.

If, by any chance, you've created a new Origin account for every game, you might want to contact a Game Advisor at https://help.ea.com/contact-us for assistance. They might be able to merge them all together into one account.