Search engines use bots, like Googlebot, to visit pages, read their content and follow the links on them. That visit is called a crawl.
A new link doesn't count until the page it's on has been crawled again after the link was added. Popular pages are crawled often, sometimes daily; small or rarely updated pages may wait weeks.
That's one reason new links can take a while to show up in backlink tools or affect rankings. It's normal, and it's not a reason to ask the editor to change anything.
Example
Margin's new link on a small blog appears in backlink tools three weeks after it went live, once the page was crawled again.
Putting it into practice
Where to use it: Nofollow link checker.