How to: Add a publish immediately button when editing custom WordPress posts?

As per the comments above: you should be able to click ‘update’ in the publish metabox.

If your post is scheduled to be published, it will be updated and still scheduled to be published at that date.

If the post is already published, it is updated (and remains published). If the post is in draft or pending status clicking ‘publish’ updates and publishes the post.