First thing to do is to refresh your account. Make sure you are logged in andckick this link:

https://www.gog.com/account/refresh

I already did that, I even tried logging out, but nothing happens.

Prime Gaming codes take a long time to get fully processed these days.
Wait 24 hours and then start worrying.

My latest PG claims took 5 to 6 hours, before they appeared in my account.
The ones before them took around 12 hours, IIRC.
Patience is still a virtue.

Unlike other GOG keys, processing of Amazon prime keys might take hours.

If you redeem multiple games in a row, some might appear instantly, others with a delay. Processing time can be different for each game.

Do not try to activate the same key more than once. Applying the same key again and again will only overload servers, and make processing longer for everyone.

If the key is about to expire, it might become too late at the time of activation attempt on the server. In that case the game will be lost.

Otherwise, if you have waited more than 24 hours, and the game has not appeared yet, try to contact GOG support.

Like others said: It can take up to 24 hours or sometimes even longer for Amazon GOG codes to appear on the accounts after they are redeemed. They will eventually appear on your library. You don't need to worry at all.