If the artist released the post to public view, are we free to upload that picture, do we have to wait for a certain time or do we have to wait for them to upload it to a free site like Ink bunny?
I'm asking because there was a point I noticed a user took two pictures from a artists pateon, that were released by the artist, but not shared yet anywhere else. The two posts however were approved by the site.

If it's a post you can access without subscription, or else the artist says you can post a certain variant freely, you can post it.

Although if the artist asks that you not repost it for a limited time, or at all, you need to obey that.

Basically anything that you need to pay to see is prohibited on the site. If the post is publicly available on an artist's official gallery (FA, IB, "Public" Patreon, etc.) and not acquired through piracy, it is okay to post it here.

For your question, as the post's type is "public" and not "pledge/patron only", the post can be freely uploaded (provided that the artist allows reposting to other sites).

Patreon is site like any other, it's just that many furry artists choose to make everything behind pledge or do so much explicit content that they are required to put them behind pledge as per patreons guidelines.

So like said above, as long as I can go to artist patreon and get the content without pledging to them, we treat is as free content.